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Exponents, Roots, and Logarithms
Laws of Exponents
An exponent tells you how many times a base number is multiplied by itself. The expression $a^n$ means $a$ multiplied by itself $n$ times. These are the fundamental rules that govern all exponent arithmetic.
$$a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}$$
An exponent is shorthand for repeated multiplication of the same factor.
$a$=The base — the number being multiplied(any real number)
$n$=The exponent (or power) — how many times the base appears(positive integer, zero, or negative integer)
$n = 1$
→$a^1 = a$ — any number to the power of 1 is itself.
$n = 0$
→$a^0 = 1$ for any $a \neq 0$ — anything (except zero) to the zero power equals 1.
$n < 0$
→$a^{-n} = \frac{1}{a^n}$ — a negative exponent flips the base into a fraction.
Base Rules: The base can be any real number — positive, negative, or a fraction. But $0^0$ is undefined, and $0^{-n}$ is also undefined.
Exponent = Count: Think of the exponent as a counter. $3^4$ means "start with 1, multiply by 3 four times": $1 \times 3 \times 3 \times 3 \times 3 = 81$.
Order Matters: $2^3 = 8$ but $3^2 = 9$. The base and exponent are NOT interchangeable.
When you multiply or divide expressions with the same base, the exponent behaves in a beautifully simple way — you just add or subtract them.
$$\frac{a^m}{a^n} = a^{m - n}$$
Same base, different exponents: multiply means add exponents, divide means subtract exponents.
$a^m \times a^n$=Product — add exponents(unitless)
$a^{m+n}$=Result of multiplying same bases(unitless)
$\frac{a^m}{a^n} = a^{m-n}$=Quotient — subtract exponents(unitless)
$m = n$
→$\frac{a^m}{a^m} = a^0 = 1$. This is why anything to the zero power equals 1.
$m < n$
→$a^{m-n}$ gives a negative exponent, meaning the result is a fraction: $\frac{1}{a^{n-m}}$.
Why It Works: $a^3 \times a^4 = (a \cdot a \cdot a)(a \cdot a \cdot a \cdot a) = a^7$. You literally just counted 7 a's total.
Division Reversal: $\frac{a^5}{a^2} = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a} = a^3$. Two a's cancel, leaving 3.
Different Bases: You CANNOT add exponents with different bases. $2^3 \times 3^2 \neq 5^5$. Convert to the same base first if possible.
When an exponentiated expression is raised to another power, you multiply the exponents. This rule also governs how exponents interact with parentheses, fractions, and negative bases.
$$(a^m)^n = a^{mn}$$
Power of a power means multiply the exponents together.
$(a^m)^n$=A base $a$ raised to $m$, then the result raised to $n$(unitless)
$mn$=Product of the two exponents(unitless)
a fraction is raised to a power
→$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ — the power distributes to both numerator and denominator.
a product is raised to a power
→$(ab)^n = a^n \cdot b^n$ — the power distributes to each factor.
Nested Powers: $((2^3)^2)^4 = 2^{3 \times 2 \times 4} = 2^{24}$. Just multiply every exponent in the chain.
Negative Base with Parentheses: $(-3)^2 = 9$ (parentheses force the negative to be included), but $-3^2 = -9$ (exponent applies to 3 only, then negate).
Fractional Power Distribution: $\left(\frac{x^2 y}{z^3}\right)^4 = \frac{x^8 y^4}{z^{12}}$ — every factor inside gets multiplied by 4.
Scientific Notation
Scientific notation is a compact way to write very large or very small numbers by expressing them as a base between 1 and 10 multiplied by a power of 10.
$$N = a \times 10^n \quad \text{where} \quad 1 \leq |a| < 10$$
Any number can be written as a coefficient (1 to 10) times a power of 10.
$a$=The coefficient — must be at least 1 and less than 10(real number)
$n$=The exponent on 10 — tells you how far the decimal point moved(integer)
writing large numbers
→Move the decimal LEFT; the exponent is positive. $45000 = 4.5 \times 10^4$
writing small numbers
→Move the decimal RIGHT; the exponent is negative. $0.0032 = 3.2 \times 10^{-3}$
Counting Places: For $45000$, move the decimal 4 places left past 4 digits → $n = 4$. For $0.0032$, move 3 places right past 3 zeros → $n = -3$.
Multiplication in Sci-Not: Multiply coefficients, add exponents: $(3 \times 10^4)(2 \times 10^3) = 6 \times 10^7$.
Division in Sci-Not: Divide coefficients, subtract exponents: $\frac{8 \times 10^6}{4 \times 10^2} = 2 \times 10^4$.
Adjusting the Coefficient: If multiplication gives a coefficient $\geq 10$, adjust: $45 \times 10^5 = 4.5 \times 10^6$.
Common Powers of 10 to Memorize
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$10^3 = 1{,}000$ (thousand)
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$10^6 = 1{,}000{,}000$ (million)
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$10^9 = 1{,}000{,}000{,}000$ (billion)
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$10^{-3} = 0.001$ (milli)
•
$10^{-6} = 0.000001$ (micro)
Roots and Surds
A root is the inverse operation of an exponent. The $n$-th root of a number $x$ is the value that, when raised to the power $n$, gives $x$.
$$\sqrt[n]{x} = x^{1/n}$$
Taking a root is the same as raising to a fractional power — the index of the root becomes the denominator of the exponent.
$\sqrt[n]{x}$=The $n$-th root of $x$(unitless)
$n$=The index of the root (2 for square root, 3 for cube root)(positive integer)
$x$=The radicand — the number under the root sign(non-negative for even roots)
$n = 2$
→The square root $\sqrt{x}$. Every positive number has two square roots: $\pm\sqrt{x}$, but $\sqrt{x}$ denotes the principal (positive) root.
$n = 3$
→The cube root $\sqrt[3]{x}$. Defined for all real numbers: $\sqrt[3]{-8} = -2$.
$n$ is even
→$\sqrt[n]{x}$ requires $x \geq 0$ (no real even roots of negative numbers).
Roots as Fractional Exponents: This is the key bridge. Every root law is just an exponent law in disguise. $\sqrt{x} = x^{1/2}$, $\sqrt[3]{x} = x^{1/3}$, $\sqrt[4]{x} = x^{1/4}$.
Multiplying Roots: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ (same index only). This lets you simplify: $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$.
Dividing Roots: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$. Works the same way — combine under one root, then simplify.
Root of a Power: $\sqrt[n]{a^m} = a^{m/n}$. The exponent inside becomes the numerator, the root index becomes the denominator.
A surd is an irrational root that cannot be simplified to a whole number — it stays under the root sign. Surds follow their own algebra, which is essential for simplification.
$$a\sqrt{n} + b\sqrt{n} = (a + b)\sqrt{n}$$
Like surds (same radicand) can be added or subtracted, just like like terms in algebra.
$a, b$=Rational coefficients (integers or fractions)(unitless)
$\sqrt{n}$=The surd part — must be identical to combine(unitless)
Simplifying Surds: Extract perfect square factors from the radicand. $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$. Always extract the largest perfect square possible.
Adding Surds: $3\sqrt{5} + 7\sqrt{5} = 10\sqrt{5}$. But $2\sqrt{3} + 4\sqrt{7}$ cannot be combined — different radicands.
Multiplying Surds: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$. For coefficients: $3\sqrt{2} \times 4\sqrt{3} = 12\sqrt{6}$.
Rationalizing Simple Surds: Multiply numerator and denominator by the surd: $\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$.
Perfect Squares to Memorize (up to 20²)
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$1, 4, 9, 16, 25, 36, 49, 64, 81, 100$
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$121, 144, 169, 196, 225, 256, 289, 324, 361, 400$
Rationalizing the denominator means eliminating all surds from the bottom of a fraction. This is a standard form requirement — answers with surds in the denominator are considered unsimplified.
$$\frac{a}{\sqrt{b} + \sqrt{c}} = \frac{a(\sqrt{b} - \sqrt{c})}{(\sqrt{b} + \sqrt{c})(\sqrt{b} - \sqrt{c})} = \frac{a(\sqrt{b} - \sqrt{c})}{b - c}$$
Multiply top and bottom by the conjugate of the denominator to eliminate surds using the difference of squares.
$\sqrt{b} + \sqrt{c}$=The original denominator (binomial surd)(unitless)
$\sqrt{b} - \sqrt{c}$=The conjugate — same terms, opposite sign between them(unitless)
$b - c$=The rational result — $(\sqrt{b})^2 - (\sqrt{c})^2$ by difference of squares(unitless)
Single Surd Denominator: Multiply by the surd itself. $\frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$.
Binomial Surd Denominator: Use the conjugate. The conjugate of $\sqrt{a} + \sqrt{b}$ is $\sqrt{a} - \sqrt{b}$, and vice versa.
Difference of Squares: $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$. This is the magic — the cross terms cancel, leaving a rational number.
Trinomial Denominators: These are rare at this level, but the technique extends by multiplying by a carefully chosen expression.
Introduction to Logarithms
A logarithm answers the question: "What exponent do I need?" If $b^y = x$, then $\log_b(x) = y$. Logarithms are the inverse operation of exponentiation — they undo what powers do.
$$\log_b(x) = y \quad \Longleftrightarrow \quad b^y = x$$
The logarithm of $x$ to the base $b$ is the exponent you raise $b$ to in order to get $x$.
$b$=The base — must be positive and not equal to 1(unitless)
$x$=The argument — must be positive ($x > 0$)(unitless)
$y$=The logarithm value (the answer) — can be any real number(unitless)
$x = 1$
→$\log_b(1) = 0$ for any base. Because $b^0 = 1$ always.
$x = b$
→$\log_b(b) = 1$. Because $b^1 = b$.
$x = b^k$
→$\log_b(b^k) = k$. The logarithm simply pulls down the exponent.
Base Restrictions: The base $b$ must satisfy $b > 0$ and $b \neq 1$. A base of 1 gives $1^y = 1$ for all $y$, so there's no unique answer.
Argument Must Be Positive: $\log_b(x)$ is undefined for $x \leq 0$. There is no real exponent that makes $b^y$ negative or zero when $b > 0$.
Reading It Aloud: "$\log_2(8)$" reads as "log base 2 of 8" and equals 3, because $2^3 = 8$.
Log Undoes Power: $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$. These are the fundamental cancellation identities.
Two specific bases appear so frequently that they have their own notation: the common logarithm (base 10) and the natural logarithm (base $e$).
$$\log(x) = \log_{10}(x), \qquad \ln(x) = \log_e(x)$$
When no base is written, $\log$ means base 10. $\ln$ always means base $e \approx 2.718$.
$\log(x)$=Common logarithm — base 10(unitless)
$\ln(x)$=Natural logarithm — base $e$(unitless)
$e$=Euler's number, approximately 2.71828(unitless)
$x = 10$
→$\log(10) = 1$ and $\log(100) = 2$ and $\log(1000) = 3$.
$x = e$
→$\ln(e) = 1$ and $\ln(e^2) = 2$.
Common Log Applications: Base-10 logs are used for pH calculations, sound intensity (decibels), and the Richter scale. Any time something spans many orders of magnitude.
Natural Log Applications: Base-$e$ logs appear in compound interest, population growth, radioactive decay, and continuous processes.
Number of Digits Trick: For any positive integer $N$, the number of digits in $N$ equals $\lfloor \log_{10}(N) \rfloor + 1$.
Useful Log Values to Know
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$\log_{10}(2) \approx 0.301$
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$\log_{10}(3) \approx 0.477$
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$\log_{10}(5) \approx 0.699$
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$\log_{10}(7) \approx 0.845$
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$\ln(2) \approx 0.693$
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$\ln(10) \approx 2.303$
Properties of Logarithms
The logarithm converts multiplication into addition and division into subtraction. These are the two most-used logarithm properties and work for any valid base.
$$\log_b(xy) = \log_b(x) + \log_b(y), \qquad \log_b\!\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$$
The log of a product is the sum of logs; the log of a quotient is the difference of logs.
$\log_b(xy)$=Log of a product — splits into a sum(unitless)
$\log_b(x/y)$=Log of a quotient — splits into a difference(unitless)
$y = 1$
→$\log_b(x) = \log_b(x) + \log_b(1) = \log_b(x) + 0$. Confirms $\log_b(1) = 0$.
$x = y$
→$\log_b(x^2) = 2\log_b(x)$. This leads to the power rule.
Why It Works: Since $b^{\log_b(x) + \log_b(y)} = b^{\log_b(x)} \cdot b^{\log_b(y)} = x \cdot y$, the exponent $\log_b(x) + \log_b(y)$ must equal $\log_b(xy)$.
Direction Matters: You can expand $\log(12) = \log(3 \times 4) = \log(3) + \log(4)$, or combine $\log(3) + \log(5) = \log(15)$. Both directions are useful.
Multiple Terms: $\log\left(\frac{ab}{c}\right) = \log(a) + \log(b) - \log(c)$. Products in the numerator add, products in the denominator subtract.
When the argument of a logarithm is raised to a power, the exponent can be brought down as a multiplier in front of the log.
$$\log_b(x^n) = n \cdot \log_b(x)$$
The exponent on the argument becomes a coefficient multiplied in front of the log.
$n$=The exponent on $x$ inside the log(any real number)
$n \cdot \log_b(x)$=The exponent pulled out as a multiplier(unitless)
$n = -1$
→$\log_b\!\left(\frac{1}{x}\right) = -\log_b(x)$. A negative exponent flips the argument into the denominator.
$n = \frac{1}{2}$
→$\log_b(\sqrt{x}) = \frac{1}{2}\log_b(x)$. Square roots become half-multipliers.
Proof by Expansion: $\log_b(x^3) = \log_b(x \cdot x \cdot x) = \log_b(x) + \log_b(x) + \log_b(x) = 3\log_b(x)$.
Expanding Complex Expressions: $\log\left(\frac{x^2 \sqrt{y}}{z^3}\right) = 2\log(x) + \frac{1}{2}\log(y) - 3\log(z)$.
Contracting (Reverse): $3\log(a) - 2\log(b) = \log\left(\frac{a^3}{b^2}\right)$. Coefficients become exponents; sums become products, differences become quotients.
The change of base formula lets you evaluate a logarithm in any base by converting it to a ratio of logs in a more convenient base — usually base 10 or base $e$.
$$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$$
To compute a log in base $b$, take the log of the argument and divide by the log of the base, both in any convenient base $c$.
$b$=The original (unwanted) base(unitless)
$c$=The new (convenient) base — typically 10 or $e$(unitless)
$x$=The argument (unchanged)(unitless)
$c = 10$
→$\log_b(x) = \frac{\log_{10}(x)}{\log_{10}(b)}$ — use when your calculator has a $\log$ button.
$c = e$
→$\log_b(x) = \frac{\ln(x)}{\ln(b)}$ — use when your calculator has an $\ln$ button.
Base Swap Identity: $\frac{\log_b(x)}{\log_b(y)} = \frac{\log_y(x)}{\log_y(y)} = \log_y(x)$. The ratio of two logs with the same base equals a single log with the denominator as the new base.
Why It Works: Let $\log_b(x) = k$, so $b^k = x$. Taking $\log_c$ of both sides: $k \cdot \log_c(b) = \log_c(x)$, giving $k = \frac{\log_c(x)}{\log_c(b)}$.
Convenient Base Choice: Pick $c$ so that $\log_c(x)$ and $\log_c(b)$ are easy. For $\log_2(8)$, you don't need this formula — but for $\log_3(50)$, use $c = 10$ with a calculator.
Solving Exponential and Logarithmic Equations
To solve an exponential equation (variable in the exponent), rewrite both sides with the same base so the exponents can be equated directly.
$$b^m = b^n \quad \Rightarrow \quad m = n$$
If two powers with the same base are equal, their exponents must be equal.
$b$=Common base on both sides (must be the same)(positive, $b \neq 1$)
$m, n$=Exponents to be equated(any real numbers)
Strategy: Convert both sides to the same base. For $2^{3x} = 32$, rewrite $32 = 2^5$, then $3x = 5$, so $x = \frac{5}{3}$.
When Bases Don't Match Directly: $9^{x+1} = 27^{2x}$. Rewrite: $(3^2)^{x+1} = (3^3)^{2x}$, so $2(x+1) = 6x$, giving $x = \frac{1}{2}$.
Using Logarithms as Fallback: If you cannot match bases, take the log of both sides: $3^x = 20 \Rightarrow x \cdot \ln(3) = \ln(20) \Rightarrow x = \frac{\ln(20)}{\ln(3)}$.
Exponential Inequalities: For $b > 1$, the inequality direction is preserved. For $0 < b < 1$, it flips. $2^x > 8 \Rightarrow x > 3$, but $(\frac{1}{2})^x > 8 \Rightarrow x < -3$.
To solve a logarithmic equation, use the logarithm properties to combine all logs into a single log on each side, then exponentiate both sides to remove the logs.
$$\log_b(x) = y \quad \Rightarrow \quad x = b^y$$
Rewrite the logarithmic equation in its equivalent exponential form to solve for the variable.
$b$=The base of the logarithm(positive, $b \neq 1$)
$x$=The argument (contains the unknown)(unitless)
$y$=The right-hand side value(unitless)
Combine First: $\log_2(x) + \log_2(x - 2) = 3$. Combine: $\log_2(x(x-2)) = 3$. Exponentiate: $x(x-2) = 8$. Solve: $x^2 - 2x - 8 = 0 \Rightarrow x = 4$ or $x = -2$. Reject $x = -2$ (argument must be positive).
Domain Check: ALWAYS verify that solutions make the argument of every log positive. A solution that gives $\log(x)$ with $x \leq 0$ is extraneous.
Quadratic in Disguise: $\log_3(x^2) - \log_3(x) = 1 \Rightarrow \log_3(x) = 1 \Rightarrow x = 3$. Note that $x > 0$ is implicit.
Application Strategy
Worked Examples
Problem Statement
Simplify $\frac{(2x^3 y^2)^4 \cdot x^5}{(2x)^3 \cdot y^6}$.
exponent
Strategic Toolkit
$$\frac{(ab)^n \cdot a^m}{(ab)^p \cdot a^q} = \frac{a^{4n+m}}{2^p \cdot a^p} \cdot \frac{b^{4n}}{b^q}$$
Step 1
Expand the numerator with the power rule
Apply the power rule to $(2x^3 y^2)^4$: each factor inside gets raised to the 4th power.
$(2x^3 y^2)^4 = 2^4 \cdot x^{12} \cdot y^8 = 16x^{12}y^8$
Step 2
Expand the denominator
Apply the power rule to $(2x)^3$.
$(2x)^3 = 2^3 \cdot x^3 = 8x^3$
Step 3
Multiply by remaining numerator factors
Multiply $16x^{12}y^8$ by the remaining $x^5$.
$16x^{12}y^8 \cdot x^5 = 16x^{17}y^8$
Step 4
Divide coefficients and subtract exponents
Divide the numerator by $8x^3 y^6$: subtract exponents for like bases, divide coefficients.
$\frac{16x^{17}y^8}{8x^3 y^6} = 2x^{14}y^2$
Problem Statement
A company's revenue is $4.2 \times 10^8$ dollars and its expenses are $3.5 \times 10^6$ dollars. What is the ratio of revenue to expenses in scientific notation?
scientific_notation
Strategic Toolkit
$$\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$$
Step 1
Set up the ratio
Write the ratio of revenue to expenses using the given scientific notation values.
$\text{Ratio} = \frac{4.2 \times 10^8}{3.5 \times 10^6}$
Step 2
Divide coefficients
Divide 4.2 by 3.5.
$\frac{4.2}{3.5} = \frac{42}{35} = 1.2$
Step 3
Subtract exponents
Subtract the exponent in the denominator from the exponent in the numerator.
$10^{8-6} = 10^2$
Step 4
Combine and express in scientific notation
The coefficient 1.2 is already between 1 and 10, so the result is valid scientific notation.
$\text{Ratio} = 1.2 \times 10^2 = 120$
Problem Statement
Simplify $2\sqrt{48} - 3\sqrt{27} + 5\sqrt{75}$.
surd
Strategic Toolkit
$$\sqrt{a^2 \cdot b} = a\sqrt{b}, \qquad a\sqrt{n} + b\sqrt{n} = (a+b)\sqrt{n}$$
Step 1
Simplify each surd individually
Extract the largest perfect square factor from each radicand: $48 = 16 \times 3$, $27 = 9 \times 3$, $75 = 25 \times 3$.
$2\sqrt{48} = 2(4\sqrt{3}) = 8\sqrt{3}, \quad 3\sqrt{27} = 3(3\sqrt{3}) = 9\sqrt{3}, \quad 5\sqrt{75} = 5(5\sqrt{3}) = 25\sqrt{3}$
Step 2
Combine like surds
All three terms now have the same radicand $\sqrt{3}$, so add and subtract the coefficients.
$8\sqrt{3} - 9\sqrt{3} + 25\sqrt{3} = (8 - 9 + 25)\sqrt{3} = 24\sqrt{3}$
Problem Statement
Simplify $\frac{6}{\sqrt{5} - \sqrt{2}}$ by rationalizing the denominator.
conjugatesurd
Strategic Toolkit
$$\frac{a}{\sqrt{b} - \sqrt{c}} = \frac{a(\sqrt{b} + \sqrt{c})}{b - c}$$
Step 1
Identify the conjugate
The denominator is $\sqrt{5} - \sqrt{2}$, so its conjugate is $\sqrt{5} + \sqrt{2}$.
$\text{Conjugate} = \sqrt{5} + \sqrt{2}$
Step 2
Multiply numerator and denominator by the conjugate
Multiply the entire fraction by $\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}}$, which equals 1.
$\frac{6}{\sqrt{5} - \sqrt{2}} \times \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}} = \frac{6(\sqrt{5} + \sqrt{2})}{(\sqrt{5})^2 - (\sqrt{2})^2}$
Step 3
Simplify the denominator
Apply the difference of squares to the denominator.
$(\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3$
Step 4
Simplify the full expression
Divide each term in the numerator by the simplified denominator 3.
$\frac{6(\sqrt{5} + \sqrt{2})}{3} = 2(\sqrt{5} + \sqrt{2}) = 2\sqrt{5} + 2\sqrt{2}$
Problem Statement
Find the value of $\log_3\left(\frac{1}{81}\right)$.
logarithm
Strategic Toolkit
$$\log_b(x) = y \quad \Longleftrightarrow \quad b^y = x$$
Step 1
Set up the exponential equation
Rewrite the logarithmic equation in its equivalent exponential form.
$\log_3\!\left(\frac{1}{81}\right) = y \quad \Longleftrightarrow \quad 3^y = \frac{1}{81}$
Step 2
Express the right side as a power of 3
Recognize that $81 = 3^4$, so $\frac{1}{81} = 3^{-4}$.
$\frac{1}{81} = \frac{1}{3^4} = 3^{-4}$
Step 3
Equate exponents
Since both sides are powers of the same base, set the exponents equal.
$3^y = 3^{-4} \quad \Rightarrow \quad y = -4$
Problem Statement
Expand $\log\!\left(\frac{x^3 \sqrt{y}}{z^2}\right)$ using logarithm properties.
logarithm
Strategic Toolkit
$$\log\!\left(\frac{A}{B}\right) = \log A - \log B, \qquad \log(x^n) = n\log x, \qquad \log(AB) = \log A + \log B$$
Step 1
Apply the quotient rule
Split the fraction into a difference of logs.
$\log\!\left(\frac{x^3 \sqrt{y}}{z^2}\right) = \log(x^3 \sqrt{y}) - \log(z^2)$
Step 2
Apply the product rule to the numerator
Split $x^3 \sqrt{y}$ into a sum of logs.
$\log(x^3 \sqrt{y}) = \log(x^3) + \log(\sqrt{y})$
Step 3
Apply the power rule to each term
Pull exponents out as coefficients. Remember $\sqrt{y} = y^{1/2}$.
$\log(x^3) = 3\log(x), \quad \log(\sqrt{y}) = \frac{1}{2}\log(y), \quad \log(z^2) = 2\log(z)$
Step 4
Write the final expanded form
Combine all the pieces together.
$3\log(x) + \frac{1}{2}\log(y) - 2\log(z)$
Problem Statement
Evaluate $\log_5(200)$ correct to two decimal places.
logarithmchange_of_base
Strategic Toolkit
$$\log_5(200) = \frac{\ln(200)}{\ln(5)}$$
Step 1
Apply the change of base formula
Convert from base 5 to base $e$ using natural logarithms.
$\log_5(200) = \frac{\ln(200)}{\ln(5)}$
Step 2
Compute the numerator
Find the natural log of 200.
$\ln(200) \approx 5.298$
Step 3
Compute the denominator
Find the natural log of 5.
$\ln(5) \approx 1.609$
Step 4
Divide and round
Divide the numerator by the denominator to get the final answer.
$\log_5(200) = \frac{5.298}{1.609} \approx 3.29$
Problem Statement
Solve for $x$: $4^{2x-1} = 8^{x+2}$.
exponent
Strategic Toolkit
$$b^m = b^n \Rightarrow m = n, \qquad (b^p)^q = b^{pq}$$
Step 1
Express both sides with the same base
Rewrite 4 and 8 as powers of 2.
$4 = 2^2, \quad 8 = 2^3$
Step 2
Apply the power rule to each side
Raise the base-2 expressions to their respective powers using $(b^p)^q = b^{pq}$.
$(2^2)^{2x-1} = 2^{2(2x-1)} = 2^{4x-2}, \quad (2^3)^{x+2} = 2^{3(x+2)} = 2^{3x+6}$
Step 3
Equate exponents
Since both sides are powers of 2 with the same base, set the exponents equal.
$4x - 2 = 3x + 6$
Step 4
Solve for x
Subtract $3x$ from both sides and add 2.
$x = 8$
Problem Statement
Solve for $x$: $\log_3(x + 5) + \log_3(x - 3) = 2$.
logarithm
Strategic Toolkit
$$\log_b(M) + \log_b(N) = \log_b(MN), \qquad \log_b(x) = y \Rightarrow x = b^y$$
Step 1
Combine the logarithms
Use the product rule to combine the two logs into one.
$\log_3((x+5)(x-3)) = 2$
Step 2
Rewrite in exponential form
Convert the log equation to its equivalent exponential equation.
$(x+5)(x-3) = 3^2 = 9$
Step 3
Expand and form a quadratic
Expand the brackets and bring all terms to one side.
$x^2 + 2x - 15 = 9 \quad \Rightarrow \quad x^2 + 2x - 24 = 0$
Step 4
Factor and solve
Factor the quadratic equation to find the two potential solutions.
$(x + 6)(x - 4) = 0 \quad \Rightarrow \quad x = -6 \text{ or } x = 4$
Step 5
Check the domain
Both $x + 5 > 0$ and $x - 3 > 0$ are required. $x = -6$ fails ($-6 - 3 < 0$), but $x = 4$ works.
$x = 4 \text{ (reject } x = -6\text{)}
Problem Statement
Write $2\log(a) - 3\log(b) + \frac{1}{2}\log(c)$ as a single logarithm.
logarithm
Strategic Toolkit
$$n\log(x) = \log(x^n), \qquad \log A - \log B = \log\!\left(\frac{A}{B}\right), \qquad \log A + \log B = \log(AB)$$
Step 1
Convert coefficients to exponents
Apply the power rule in reverse to turn each coefficient into an exponent.
$2\log(a) = \log(a^2), \quad -3\log(b) = \log(b^{-3}) = \log\!\left(\frac{1}{b^3}\right), \quad \frac{1}{2}\log(c) = \log(c^{1/2}) = \log(\sqrt{c})$
Step 2
Combine the first two terms
Use the quotient rule to combine $\log(a^2)$ and $\log(b^{-3})$.
$\log(a^2) - \log(b^3) = \log\!\left(\frac{a^2}{b^3}\right)$
Step 3
Add the third term
Use the product rule to incorporate $\log(\sqrt{c})$.
$\log\!\left(\frac{a^2}{b^3}\right) + \log(\sqrt{c}) = \log\!\left(\frac{a^2 \sqrt{c}}{b^3}\right)$
Memorize formulas
Product Rule
When multiplying same bases, add the exponents
Formula
$a^m \times a^n = a^{m+n}$
Quotient Rule
When dividing same bases, subtract the exponents
Formula
$\frac{a^m}{a^n} = a^{m-n}$
Power Rule
When raising a power to another power, multiply the exponents
Formula
$(a^m)^n = a^{mn}$
Zero Exponent
Any non-zero number to the zero power equals 1
Formula
$a^0 = 1 \quad (a \neq 0)$
Negative Exponent
A negative exponent means reciprocal
Formula
$a^{-n} = \frac{1}{a^n}$
Fractional Exponent (Root)
The numerator is the power, the denominator is the root
Formula
$a^{m/n} = \sqrt[n]{a^m}$
Logarithm Definition
The log is the exponent needed
Formula
$\log_b(x) = y \Leftrightarrow b^y = x$
Log Product Rule
Log of a product equals sum of logs
Formula
$\log_b(xy) = \log_b(x) + \log_b(y)$
Log Quotient Rule
Log of a quotient equals difference of logs
Formula
$\log_b\!\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$
Log Power Rule
Pull the exponent down as a multiplier
Formula
$\log_b(x^n) = n \cdot \log_b(x)$
Change of Base
Convert a log in any base to a ratio of logs in a convenient base
Formula
$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$
Difference of Squares (Rationalizing)
Multiply by the conjugate to eliminate surds from the denominator
Formula
$(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$
Review
1
A population of bacteria doubles every 2 hours. Starting from $2^5$ bacteria, how many are there after 6 hours?
Ans
256
Review
Tap to reveal
Determine the number of doublings
Each doubling occurs every 2 hours, so 6 hours gives $6 \div 2 = 3$ doublings.
$$3 \text{ doublings}$$
Apply the product rule
Each doubling multiplies by $2^1$, so three doublings multiply by $2^3$. Use $a^m \times a^n = a^{m+n}$.
$$2^5 \times 2^3 = 2^8 = 256$$
2
Simplify $\frac{3^7 \times 3^{-2}}{3^3}$.
Ans
9
Review
Tap to reveal
Combine exponents using the product and quotient rules
Apply $a^m \times a^n = a^{m+n}$ to the numerator, then $\frac{a^m}{a^n} = a^{m-n}$ to divide.
$$\frac{3^{7-2}}{3^3} = \frac{3^5}{3^3} = 3^{5-3} = 3^2$$
Evaluate
Compute $3^2$.
$$3^2 = 9$$
3
The distance from Earth to the Sun is approximately $1.5 \times 10^{11}$ meters. If light travels at $3 \times 10^8$ m/s, how many seconds does light take to reach Earth?
Ans
500
Review
Tap to reveal
Set up the division in scientific notation
Divide the distance by the speed of light, separating coefficients and powers of 10.
$$\frac{1.5 \times 10^{11}}{3 \times 10^8} = \frac{1.5}{3} \times \frac{10^{11}}{10^8}$$
Simplify using the quotient rule
Apply $\frac{10^{11}}{10^8} = 10^{11-8} = 10^3$ and divide the coefficients.
$$0.5 \times 10^3 = 500 \text{ seconds}$$
4
If $\log_5(x) = 3$, what is the value of $x$?
Ans
125
Review
Tap to reveal
Apply the logarithm definition
Convert the logarithmic statement to its equivalent exponential form using $\log_b(x) = y \Leftrightarrow b^y = x$.
$$x = 5^3$$
Evaluate the power
Compute $5^3$.
$$x = 125$$
5
Simplify $(2^3)^2 \times 2^{-4}$.
Ans
4
Review
Tap to reveal
Apply the power rule
Use $(a^m)^n = a^{mn}$ to simplify the grouped power.
$$(2^3)^2 = 2^{3 \times 2} = 2^6$$
Combine with the negative exponent
Use $a^m \times a^n = a^{m+n}$ to merge the exponents.
$$2^6 \times 2^{-4} = 2^{6-4} = 2^2 = 4$$
6
A radioactive substance decays to $(\frac{1}{2})^n$ of its original mass after $n$ half-lives. If the original mass is $4^3$ grams, what is the mass after 4 half-lives?
Ans
4
Review
Tap to reveal
Express the remaining fraction
After 4 half-lives, the fraction remaining is $(\frac{1}{2})^4 = \frac{1}{16}$.
$$(\frac{1}{2})^4 = \frac{1}{16}$$
Multiply by the original mass
Compute $4^3 \times \frac{1}{16} = 64 \times \frac{1}{16}$.
$$4^3 \times \frac{1}{16} = 4 \text{ grams}$$
7
The volume of a cube-shaped container is expressed as $x^6 y^{12}$ cubic units. What is the side length of the container?
Ans
$x^2 y^4$
Review
Tap to reveal
Set up the cube root
For a cube with volume $V$, the side length is $V^{1/3}$.
$$s = (x^6 y^{12})^{1/3}$$
Apply the fractional exponent rule
Use $a^{m/n} = (a^m)^{1/n}$ to distribute the exponent $\frac{1}{3}$ to each factor.
$$s = x^{6 \cdot \frac{1}{3}} \cdot y^{12 \cdot \frac{1}{3}} = x^2 y^4$$
8
Simplify $\sqrt{50} + \sqrt{8}$ and express the result in simplest surd form.
Ans
$7\sqrt{2}$
Review
Tap to reveal
Simplify each surd separately
Factor each radicand to extract perfect square factors.
$$\sqrt{50} = 5\sqrt{2}, \qquad \sqrt{8} = 2\sqrt{2}$$
Combine like surds
Add the coefficients since both terms share the same irrational part $\sqrt{2}$.
$$5\sqrt{2} + 2\sqrt{2} = 7\sqrt{2}$$
9
A rectangular field has an area of $\sqrt{48} \times \sqrt{27}$ square meters. What is the area in simplest form?
Ans
36
Review
Tap to reveal
Combine under a single radical
Use $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ to multiply the two surds.
$$\sqrt{48} \times \sqrt{27} = \sqrt{48 \times 27}$$
Evaluate the product under the radical
Compute $48 \times 27 = 1296$, then find $\sqrt{1296}$.
$$\sqrt{1296} = 36$$
10
Expand $\log_3\!\left(\frac{27x^2}{y^5}\right)$ using the properties of logarithms.
Ans
$3 + 2\log_3(x) - 5\log_3(y)$
Review
Tap to reveal
Apply the quotient and product rules
Split $\log_3\!\left(\frac{27x^2}{y^5}\right)$ into separate logarithms using $\log_b\!\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$ and $\log_b(xy) = \log_b(x) + \log_b(y)$.
$$\log_3(27) + \log_3(x^2) - \log_3(y^5)$$
Evaluate and apply the power rule
Compute $\log_3(27) = 3$ and use $\log_b(x^n) = n \cdot \log_b(x)$ on the remaining terms.
$$3 + 2\log_3(x) - 5\log_3(y)$$
11
Rationalize the denominator of $\frac{6}{\sqrt{7} - \sqrt{3}}$.
Ans
$\frac{3(\sqrt{7} + \sqrt{3})}{2}$
Review
Tap to reveal
Multiply by the conjugate
Multiply both numerator and denominator by $\sqrt{7} + \sqrt{3}$ using the difference of squares formula.
$$\frac{6(\sqrt{7} + \sqrt{3})}{(\sqrt{7})^2 - (\sqrt{3})^2} = \frac{6(\sqrt{7} + \sqrt{3})}{7 - 3}$$
Simplify the fraction
Reduce $\frac{6}{4}$ to $\frac{3}{2}$.
$$\frac{3(\sqrt{7} + \sqrt{3})}{2}$$
12
A cell culture grows such that the number of cells after $t$ hours is given by $N = 4^{t+1}$. If the lab counts 64 cells, how many hours have passed?
Ans
2
Review
Tap to reveal
Express 64 as a power of 4
Rewrite 64 in terms of the base 4 to match the growth formula.
$$64 = 4^3$$
Equate the exponents
Since the bases are the same, set $t + 1 = 3$ and solve for $t$.
$$t + 1 = 3 \implies t = 2 \text{ hours}$$
13
A car depreciates according to $V = V_0 \times (0.5)^t$, where $t$ is in years. After how many years will the car's value be $\frac{1}{16}$ of its original value?
Ans
4
Review
Tap to reveal
Set up the equation
The car is worth $\frac{V_0}{16}$, so $(0.5)^t = \frac{1}{16}$.
$$(0.5)^t = \frac{1}{16}$$
Express both sides with the same base
Write $0.5$ and $\frac{1}{16}$ as powers of $\frac{1}{2}$.
$$(\frac{1}{2})^t = (\frac{1}{2})^4 \implies t = 4$$
14
Evaluate $\log_8(32)$ using the change of base formula with base 2.
Ans
$\frac{5}{3}$
Review
Tap to reveal
Apply the change of base formula
Use $\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$ with $c = 2$.
$$\log_8(32) = \frac{\log_2(32)}{\log_2(8)}$$
Evaluate the base-2 logarithms
Since $32 = 2^5$ and $8 = 2^3$, substitute and simplify.
$$\frac{5}{3}$$
15
Simplify $\frac{8^{2/3} \times 27^{1/3}}{9^{1/2}}$.
Ans
4
Review
Tap to reveal
Evaluate each fractional power
Convert each base to a prime power and apply the fractional exponent rule $a^{m/n} = (a^{1/n})^m$.
$$8^{2/3} = 4, \quad 27^{1/3} = 3, \quad 9^{1/2} = 3$$
Compute the final value
Substitute the evaluated values into the expression.
$$\frac{4 \times 3}{3} = 4$$
16
If $x = 2^{1/3} + 2^{-1/3}$, what is the value of $x^3 - 3x$?
Ans
$\frac{5}{2}$
Review
Tap to reveal
Expand $x^3$ using the binomial formula
Apply $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ with $a = 2^{1/3}$ and $b = 2^{-1/3}$.
$$x^3 = 2 + \frac{1}{2} + 3 \cdot 2^{1/3} + 3 \cdot 2^{-1/3}$$
Group terms and simplify
Recognize that $3 \cdot 2^{1/3} + 3 \cdot 2^{-1/3} = 3(2^{1/3} + 2^{-1/3}) = 3x$.
$$x^3 = \frac{5}{2} + 3x \implies x^3 - 3x = \frac{5}{2}$$
17
Solve $\log_5(x) + \log_5(x - 4) = 1$.
Ans
5
Review
Tap to reveal
Combine the logarithms
Use $\log_b(x) + \log_b(y) = \log_b(xy)$ to form a single logarithm.
$$\log_5(x(x - 4)) = 1$$
Convert to a quadratic equation
Rewrite in exponential form and expand: $x(x-4) = 5$.
$$x^2 - 4x - 5 = 0 \implies (x-5)(x+1) = 0$$
Check the domain
Reject $x = -1$ (logarithm undefined for negative arguments) and $x = 4$ (logarithm undefined at zero). The only valid solution is $x = 5$.
$$x = 5$$
18
The decibel scale measures sound intensity logarithmically. A whisper produces sound at 20 dB and normal conversation at 60 dB. The intensity ratio between conversation and whisper is given by $10^{(60-20)/10}$. What is this ratio?
Ans
$10000$
Review
Tap to reveal
Calculate the exponent
Subtract the decibel values and divide by 10 to get the power of 10.
$$\frac{60 - 20}{10} = 4$$
Evaluate $10^4$
Compute the power of 10 to find the intensity ratio.
$$10^4 = 10000$$
19
If $\log_a(2) = m$ and $\log_a(5) = n$, express $\log_a(0.16)$ in terms of $m$ and $n$.
Ans
$2m - 2n$
Review
Tap to reveal
Express 0.16 using bases 2 and 5
Rewrite the decimal as a fraction of powers of 2 and 5.
$$0.16 = \frac{4}{25} = \frac{2^2}{5^2}$$
Apply logarithm properties
Use the quotient rule and power rule to express in terms of $m$ and $n$.
$$\log_a(0.16) = 2\log_a(2) - 2\log_a(5) = 2m - 2n$$
20
If $5^{2x-1} = 125^{-(x-2)}$, find the value of $x$.
Ans
$\frac{7}{5}$
Review
Tap to reveal
Express the right side with base 5
Rewrite $125$ as $5^3$ and distribute the negative exponent.
$$125^{-(x-2)} = 5^{-3(x-2)} = 5^{-3x+6}$$
Equate the exponents
Since the bases are equal, set the exponents equal and solve for $x$.
$$2x - 1 = -3x + 6 \implies 5x = 7 \implies x = \frac{7}{5}$$
Question 1 of 20
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1
A population of bacteria doubles every 2 hours. Starting from $2^5$ bacteria, how many are there after 6 hours?