Critical Logarithmic Function Rules

Why This Matters

Logarithmic functions are the inverse of exponential functions, and understanding this relationship is one of the most powerful tools you'll use in AP Precalculus. You're being tested on your ability to move fluently between exponential and logarithmic forms, manipulate complex expressions using log rules, and recognize domain restrictions that affect solutions. These skills show up everywhere—from solving equations to analyzing function behavior to modeling real-world phenomena like population growth, sound intensity, and pH levels.

Don't just memorize these rules as isolated formulas. Each rule reflects a deeper principle about how logarithms translate multiplicative relationships into additive ones. When you understand why the product rule turns multiplication into addition, you'll never forget it—and you'll spot opportunities to apply it on exam day. Know what concept each rule illustrates, and you'll handle any logarithm problem they throw at you.

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Foundational Definitions

Before you can manipulate logarithms, you need rock-solid understanding of what a logarithm actually is. The definition creates a bridge between exponential and logarithmic forms—master this translation, and equation-solving becomes straightforward.

Definition of Logarithm

• $\log_a(x) = y$ means $a^y = x$—the logarithm answers "what exponent gives me x?"
• Base $a$ must be positive and not equal to 1—these restrictions ensure the function behaves properly
• Converting between forms is the most fundamental skill for solving logarithmic equations

The Natural Logarithm

• $\ln(x) = \log_e(x)$ where $e \approx 2.718$—Euler's number appears throughout calculus
• Growth and decay models use natural logarithms because $e$ has special derivative properties
• Your calculator's "ln" button computes this specific logarithm, making it essential for applications

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Domain and Range Restrictions

Understanding where logarithmic functions exist and what values they can produce is critical for graphing, solving equations, and checking solutions. These restrictions come directly from the definition—you can't raise a positive base to any power and get zero or a negative number.

Domain: $x > 0$

• Logarithms only accept positive inputs—you cannot take $\log_a(0)$ or $\log_a(\text{negative})$
• Always check solutions by substituting back; reject any that make the argument non-positive
• Graphically, the function has a vertical asymptote at $x = 0$

Range: All Real Numbers

• Output can be any real number—positive, negative, or zero
• As $x \to 0^+$, the logarithm approaches $-\infty$ (vertical asymptote behavior)
• As $x \to \infty$, the logarithm increases without bound, but very slowly

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Logarithm Manipulation Rules

These three rules—product, quotient, and power—are your workhorses for simplifying and solving. They all stem from the same principle: logarithms convert multiplicative operations into additive ones. This is why logarithms were invented for computation before calculators existed.

Product Rule

• $\log_a(xy) = \log_a(x) + \log_a(y)$—multiplication inside becomes addition outside
• Expanding expressions helps when solving equations with multiple log terms
• Works because $a^m \cdot a^n = a^{m+n}$—the log rule mirrors the exponent rule

Quotient Rule

• $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$—division inside becomes subtraction outside
• Useful for condensing two separate log terms into one logarithm
• Mirrors the exponent rule $\frac{a^m}{a^n} = a^{m-n}$

Power Rule

• $\log_a(x^n) = n \cdot \log_a(x)$—exponents inside become coefficients outside
• Essential for solving equations where the variable is in an exponent
• Works in reverse too: $3\log_a(x) = \log_a(x^3)$

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Inverse Relationships

Logarithmic and exponential functions undo each other. This inverse relationship is the key to solving equations—apply the inverse operation to isolate your variable.

Inverse Properties

• $\log_a(a^x) = x$—taking the log "cancels" the exponential with matching base
• $a^{\log_a(x)} = x$—exponentiating "cancels" the logarithm with matching base
• Bases must match for cancellation to work; otherwise, use change of base

Change of Base Formula

• $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$—converts any log to a different base
• Calculator application: use $\log_a(x) = \frac{\ln(x)}{\ln(a)}$ or $\frac{\log(x)}{\log(a)}$
• Proves that all logarithms differ only by a constant multiple—they're all proportional

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Quick Reference Table

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Self-Check Questions

1. Which two rules would you use together to fully expand $\log_2\left(\frac{x^3y}{z}\right)$?

2. A student solves $\log(x-3) + \log(x) = 1$ and gets $x = 5$ and $x = -2$. Which solution is valid, and why must the other be rejected?

3. Compare and contrast the product rule and power rule: when would you use each, and what common error do students make when choosing between them?

4. If you need to evaluate $\log_5(12)$ on a calculator that only has $\log$ and $\ln$ buttons, what expression would you enter?

5. Explain why the domain of $f(x) = \log_3(x-2)$ is $x > 2$ rather than $x > 0$, and describe how this affects the graph compared to $g(x) = \log_3(x)$.