Chapter Review

Functions and Logarithms

Exponents, Roots, and Logarithms · Functions Basics

Laws of Exponents

Exponent rules govern how powers combine under multiplication, division, and nesting. Every exponent manipulation reduces to three core laws plus the zero and negative exponent definitions.

Key Points

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Product Rule: Same base, multiply → add exponents: $a^m \times a^n = a^{m+n}$
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Quotient Rule: Same base, divide → subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
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Power Rule: Power of a power → multiply exponents: $(a^m)^n = a^{mn}$
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Zero Exponent: $a^0 = 1$ for any $a \neq 0$
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Negative Exponent: $a^{-n} = \frac{1}{a^n}$ — flip into a fraction
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Different Bases: Cannot combine exponents across different bases — convert to a common base first
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Power distributes over products and fractions: $(ab)^n = a^n b^n$, $(\frac{a}{b})^n = \frac{a^n}{b^n}$

Formula

$$a^m \times a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}$$

Scientific Notation

Scientific notation expresses any number as a coefficient between 1 and 10 times a power of 10, making arithmetic with very large or small numbers tractable.

Key Points

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Standard Form: $N = a \times 10^n$ where $1 \leq |a| < 10$
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Large Numbers: Decimal moves left → positive exponent. $45000 = 4.5 \times 10^4$
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Small Numbers: Decimal moves right → negative exponent. $0.0032 = 3.2 \times 10^{-3}$
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Multiply: Multiply coefficients, add exponents: $(3 \times 10^4)(2 \times 10^3) = 6 \times 10^7$
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Divide: Divide coefficients, subtract exponents: $\frac{8 \times 10^6}{4 \times 10^2} = 2 \times 10^4$
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If the product coefficient is $\geq 10$, re-adjust: $45 \times 10^5 = 4.5 \times 10^6$

Roots and Surds

Roots are the inverse of exponents, expressed as fractional powers. Surds are irrational roots that follow specific algebraic rules for simplification, combination, and rationalization.

Key Points

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Fractional Exponent Bridge: $\sqrt[n]{x} = x^{1/n}$, so every root law is an exponent law in disguise
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Multiplying Roots: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ (same index only)
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Simplifying Surds: Extract the largest perfect square: $\sqrt{50} = 5\sqrt{2}$
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Like Surds Add: $a\sqrt{n} + b\sqrt{n} = (a+b)\sqrt{n}$; different radicands cannot combine
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$\sqrt{x^2} = |x|$, not $x$ — the principal root is always non-negative
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$\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}$ — the root of a sum is not the sum of the roots

Formula

$$\sqrt[n]{x} = x^{1/n}, \quad \sqrt[n]{a^m} = a^{m/n}$$

Rationalizing Denominators

Rationalizing eliminates surds from the denominator by multiplying by the conjugate (for binomial surds) or the surd itself (for single surds), producing a simplified standard form.

Key Points

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Single Surd: Multiply by the surd: $\frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$
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Binomial Surd: Multiply by the conjugate — same terms, opposite middle sign
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Difference of Squares: $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$ eliminates the surds
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Always multiply the entire fraction (numerator and denominator) by the same conjugate factor
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Simplify the numerator fully after rationalizing — expand, combine like terms, factor

Formula

$$\frac{a}{\sqrt{b} + \sqrt{c}} = \frac{a(\sqrt{b} - \sqrt{c})}{b - c}$$

Logarithm Definition and Properties

Logarithms are the inverse of exponentiation — they answer 'what exponent do I need?' The three log properties (product, quotient, power) mirror the exponent laws and are the backbone of all logarithmic algebra.

Key Points

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Definition: $\log_b(x) = y \iff b^y = x$ — base $b$ must be positive and $\neq 1$, argument $x > 0$
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Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$ — log of product = sum of logs
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Quotient Rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$ — log of quotient = difference of logs
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Power Rule: $\log_b(x^n) = n \cdot \log_b(x)$ — bring the exponent down as a multiplier
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Cancellation: $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$
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No rule exists for $\log(x + y)$ — the log of a sum cannot be split
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Common log ($\log$) = base 10; natural log ($\ln$) = base $e \approx 2.718$

Formula

$$\log_b(xy) = \log_b(x) + \log_b(y), \quad \log_b\!\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y), \quad \log_b(x^n) = n \cdot \log_b(x)$$

Change of Base and Solving Logarithmic Equations

The change of base formula converts any logarithm into a ratio of logs in a convenient base. Logarithmic equations are solved by combining logs into a single expression, then exponentiating — always checking the domain for extraneous solutions.

Key Points

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Change of Base: $\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$ — argument on top, base on bottom
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Solving Log Equations: Combine all logs into one per side, then rewrite in exponential form
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Domain Check: Every solution must make all log arguments positive — reject any that don't
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Quadratic-in-disguise pattern: log equations often produce a quadratic, with exactly one root failing the domain check
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Chain identity: $\log_a(b) \cdot \log_b(c) = \log_a(c)$

Formula

$$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$$

Solving Exponential Equations

Exponential equations place the variable in the exponent. The primary strategy is rewriting both sides with the same base so exponents can be equated directly; logarithms serve as a fallback when bases cannot be matched.

Key Points

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Same-Base Strategy: Rewrite both sides as powers of a common base, then equate exponents: $b^m = b^n \Rightarrow m = n$
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Example: $9^{x+1} = 27^{2x}$ → $(3^2)^{x+1} = (3^3)^{2x}$ → $2(x+1) = 6x$
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Log Fallback: If bases can't match, take log of both sides: $3^x = 20 \Rightarrow x = \frac{\ln 20}{\ln 3}$
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Inequalities: For $b > 1$, inequality direction preserved; for $0 < b < 1$, it flips
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Always verify the solution by substituting back into the original equation

Formula

$$b^m = b^n \quad \Rightarrow \quad m = n$$

Function Fundamentals

A function maps each valid input to exactly one output. Understanding domain (valid inputs), range (possible outputs), and evaluation (substitution) is the foundation for all function-based problem solving.

Key Points

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Unique Output: Each input produces exactly one output — this distinguishes functions from general relations
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Domain Restrictions: Denominators $\neq 0$, square-root radicands $\geq 0$, log arguments $> 0$
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Range: The set of all actual outputs — a subset of the codomain, dependent on domain
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Notation: $f(x)$ means evaluation at $x$, NOT $f \times x$
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Evaluation: Replace every instance of the variable with the given value or expression, then simplify

Linear and Quadratic Functions

Linear functions produce constant-rate-of-change relationships (straight lines), while quadratic functions produce accelerating or decelerating change (parabolas). The vertex formula finds the maximum or minimum of any quadratic.

Key Points

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Linear: $f(x) = mx + b$ — slope $m$ is the constant rate of change, $b$ is the y-intercept
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Slope from Two Points: $m = \frac{y_2 - y_1}{x_2 - x_1}$
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Quadratic: $f(x) = ax^2 + bx + c$ — $a > 0$ opens up (minimum), $a < 0$ opens down (maximum)
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Vertex: $x_v = -\frac{b}{2a}$ gives the input at the turning point; $f(x_v)$ is the max or min output
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Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$ mirrors the parabola
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Range of Quadratic: If $a > 0$, range is $[f(x_v), \infty)$; if $a < 0$, range is $(-\infty, f(x_v)]$

Formula

$$x_v = -\frac{b}{2a}$$

Piecewise Functions and Composition

Piecewise functions apply different rules to different parts of their domain, modeling tiered real-world scenarios. Composition chains functions together — the output of one becomes the input of the next — and order matters.

Key Points

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Piecewise: Different formulas for different domain intervals — check which condition applies before evaluating
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Composition: $(f \circ g)(x) = f(g(x))$ — compute inner function $g(x)$ first, then feed result into $f$
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Order Matters: $f(g(x)) \neq g(f(x))$ in general — composition is not commutative
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Domain of Composition: All $x$ in domain of $g$ such that $g(x)$ is in domain of $f$
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Decomposition: To split $h(x) = (2x+1)^2$ into $f(g(x))$, set inner $g(x) = 2x+1$ and outer $f(x) = x^2$

Formula

$$(f \circ g)(x) = f(g(x))$$

Formulas

Exponent Power Rule

Power of a power — multiply the exponents

Formula

$(a^m)^n = a^{mn}$

Logarithm Definition

Log is the inverse of exponentiation — it recovers the exponent

Formula

$\log_b(x) = y \Leftrightarrow b^y = x$

Log Product Rule

Log of a product splits into a sum of logs

Formula

$\log_b(xy) = \log_b(x) + \log_b(y)$

Log Power Rule

Bring the exponent down as a coefficient

Formula

$\log_b(x^n) = n \cdot \log_b(x)$

Change of Base Formula

Convert any log to a ratio of logs in a convenient base

Formula

$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$

Vertex Formula

X-coordinate of the parabola's turning point (max or min)

Formula

$x_v = -\frac{b}{2a}$

Linear Function

Constant rate of change — slope times input plus intercept

Formula

$f(x) = mx + b$

Rationalizing with Conjugate

Multiply by conjugate to eliminate binomial surd denominator

Formula

$(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$