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)

$n = 1$

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$a^1 = a$ — any number to the power of 1 is itself.

$n = 0$

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$a^0 = 1$ for any $a \neq 0$ — anything (except zero) to the zero power equals 1.

$n < 0$

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$a^{-n} = \frac{1}{a^n}$ — 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)

$m = n$

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$\frac{a^m}{a^m} = a^0 = 1$. This is why anything to the zero power equals 1.

$m < n$

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$a^{m-n}$ 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

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$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ — the power distributes to both numerator and denominator.

a product is raised to a power

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$(ab)^n = a^n \cdot b^n$ — 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

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Move the decimal LEFT; the exponent is positive. $45000 = 4.5 \times 10^4$

writing small numbers

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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)

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$10^{-6} = 0.000001$ (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)

$n = 2$

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The square root $\sqrt{x}$. Every positive number has two square roots: $\pm\sqrt{x}$, but $\sqrt{x}$ denotes the principal (positive) root.

$n = 3$

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The cube root $\sqrt[3]{x}$. Defined for all real numbers: $\sqrt[3]{-8} = -2$.

$n$ is even

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$\sqrt[n]{x}$ 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)

$x = 1$

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$\log_b(1) = 0$ for any base. Because $b^0 = 1$ always.

$x = b$

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$\log_b(b) = 1$. Because $b^1 = b$.

$x = b^k$

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$\log_b(b^k) = k$. 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)

$x = 10$

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$\log(10) = 1$ and $\log(100) = 2$ and $\log(1000) = 3$.

$x = e$

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$\ln(e) = 1$ 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)

$y = 1$

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$\log_b(x) = \log_b(x) + \log_b(1) = \log_b(x) + 0$. Confirms $\log_b(1) = 0$.

$x = y$

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$\log_b(x^2) = 2\log_b(x)$. 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)

$n = -1$

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$\log_b\!\left(\frac{1}{x}\right) = -\log_b(x)$. A negative exponent flips the argument into the denominator.

$n = \frac{1}{2}$

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$\log_b(\sqrt{x}) = \frac{1}{2}\log_b(x)$. 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)

$c = 10$

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$\log_b(x) = \frac{\log_{10}(x)}{\log_{10}(b)}$ — use when your calculator has a $\log$ button.

$c = e$

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$\log_b(x) = \frac{\ln(x)}{\ln(b)}$ — 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.