Logarithmic Function Properties

Why This Matters

Logarithms aren't just another function to memorize—they're the key to unlocking exponential equations that would otherwise be unsolvable. Every time you need to isolate a variable trapped in an exponent, logarithms come to the rescue. In Honors Pre-Calc, you're being tested on your ability to manipulate logarithmic expressions fluently, convert between exponential and logarithmic forms, and recognize how logarithmic graphs behave. These skills directly feed into calculus, where logarithmic differentiation and integration become essential tools.

The properties you'll learn here—product, quotient, and power rules—aren't random formulas. They're direct consequences of how exponents work, just viewed from a different angle. Understanding this connection means you can derive any property you forget during an exam. Don't just memorize that $\log_b(xy) = \log_b(x) + \log_b(y)$—know why multiplication becomes addition when you're working with exponents. That conceptual understanding is what separates students who struggle from those who thrive.

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Foundational Definitions

Before diving into properties, you need rock-solid understanding of what a logarithm actually is. Think of it as asking the inverse question: instead of "what do I get when I raise this base to this power?" you're asking "what power gives me this result?"

Definition of Logarithms

• A logarithm answers: "What exponent produces this result?"—if $b^x = y$, then $x = \log_b(y)$
• Domain restriction: only defined for positive arguments $y > 0$ and positive bases $b > 0$ where $b \neq 1$
• The base-exponent-result relationship is the foundation for converting between exponential and logarithmic forms on every exam problem

Inverse Relationship with Exponentials

• Logarithmic and exponential functions are inverses—their graphs reflect across the line $y = x$
• Composition property: $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$ cancel each other out
• Equation-solving power: this inverse relationship lets you isolate variables trapped in exponents or inside logarithms

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Identity Properties

These two properties seem simple, but they're your anchors when simplifying complex expressions. They come directly from basic exponent facts: anything to the zero power is 1, and anything to the first power is itself.

Logarithm of 1

• $\log_b(1) = 0$ for any valid base—because $b^0 = 1$ always
• Simplification shortcut: any term containing $\log(1)$ immediately becomes zero
• Works for all bases including common log ($\log$), natural log ($\ln$), and any $\log_b$

Logarithm of the Base

• $\log_b(b) = 1$ always—because $b^1 = b$ by definition
• Quick mental check: $\log_{10}(10) = 1$, $\ln(e) = 1$, $\log_2(2) = 1$
• Useful for rewriting expressions: recognizing when an argument equals the base lets you substitute 1 instantly

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Expansion and Condensing Rules

These three rules are the workhorses of logarithm manipulation. They let you expand a single logarithm into multiple terms (useful for solving) or condense multiple logarithms into one (useful for simplifying). They work because logarithms convert multiplication/division into addition/subtraction.

Product Rule

• $\log_b(xy) = \log_b(x) + \log_b(y)$—multiplication inside becomes addition outside
• Why it works: if $b^m = x$ and $b^n = y$, then $b^{m+n} = xy$, so the exponents add
• Exam application: expand $\log_3(5x)$ as $\log_3(5) + \log_3(x)$ to isolate variables

Quotient Rule

• $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$—division inside becomes subtraction outside
• Parallel to exponents: when you divide powers with the same base, you subtract exponents
• Common mistake: students write $\frac{\log_b(x)}{\log_b(y)}$ instead—that's not the quotient rule

Power Rule

• $\log_b(x^k) = k \cdot \log_b(x)$—exponents inside become coefficients outside
• Most powerful for solving: this rule lets you "bring down" exponents to isolate variables
• Works for any real exponent $k$, including fractions and negatives

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Converting Between Bases

When your calculator only has $\log$ (base 10) and $\ln$ (base $e$), how do you evaluate $\log_5(17)$? The change of base formula bridges this gap and appears constantly in both computational and proof-based problems.

Change of Base Formula

• $\log_b(a) = \frac{\log_k(a)}{\log_k(b)}$ for any positive base $k$—typically you'll use $k = 10$ or $k = e$
• Calculator strategy: $\log_5(17) = \frac{\ln(17)}{\ln(5)} = \frac{\log(17)}{\log(5)} \approx 1.76$
• Proof technique: this formula is essential when you need to show two logarithmic expressions are equal

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Graphical Behavior

Understanding the graph of $y = \log_b(x)$ helps you visualize domain restrictions, predict function behavior, and connect algebraic properties to geometric features.

Domain and Range

• Domain: $x > 0$ only—you cannot take the logarithm of zero or negative numbers
• Range: all real numbers $(-\infty, \infty)$—logarithmic outputs can be any value
• Implication for solving: when you get a solution, always verify the argument stays positive

Graph Characteristics

• Passes through $(1, 0)$ and $(b, 1)$—these anchor points come directly from identity properties
• Vertical asymptote at $x = 0$—the graph approaches but never touches the y-axis
• Base determines direction: increasing for $b > 1$, decreasing for $0 < b < 1$

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Quick Reference Table

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Self-Check Questions

1. Which two properties allow you to immediately simplify $\log_7(1) + \log_7(7)$ without a calculator, and what is the result?

2. A student claims that $\log_4(16) - \log_4(2) = \log_4(8)$. Use the quotient rule to verify whether this is correct, and explain your reasoning.

3. Compare and contrast the change of base formula with the quotient rule—why does $\frac{\log(12)}{\log(3)}$ equal $\log_3(12)$ but $\log\left(\frac{12}{3}\right)$ does not?

4. If you're asked to solve $5^{2x+1} = 80$ on an FRQ, which logarithm property would you use first to isolate $x$, and why?

5. The graph of $y = \log_b(x)$ passes through $(9, 2)$. What is the base $b$, and how does this point relate to the definition of logarithms?