Common Logarithm Rules

Why This Matters

Logarithm rules aren't just algebraic tricks—they're the key to unlocking exponential equations that would otherwise be impossible to solve. In Precalculus, you're being tested on your ability to manipulate, simplify, and solve using these rules, which means understanding when and why to apply each one. These same principles show up everywhere from calculating earthquake magnitudes to modeling population growth to understanding sound intensity.

Here's the thing: the AP exam and your course assessments won't just ask you to state a rule. They'll give you a messy expression and expect you to recognize which rule transforms it into something workable. Don't just memorize formulas—know what each rule does to an expression and when it's your best tool. Master the underlying logic, and you'll be able to derive any rule you temporarily forget.

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The Fundamental Definition

Before diving into rules, you need the definition that makes everything else click. A logarithm answers the question: "What exponent gives me this result?"

Definition of Common Logarithm

• $\log(x) = y$ means $10^y = x$—the logarithm is the exponent that produces x when applied to base 10
• Common logarithms use base 10 by default, which is why no subscript appears (when you see $\log$ without a base, assume 10)
• This definition is your decoder ring—every rule below derives from translating between logarithmic and exponential forms

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Rules That Break Apart Expressions

These rules let you decompose complicated logarithmic expressions into simpler pieces. They work because logarithms convert multiplication and division into addition and subtraction—the same way exponents do.

Product Rule

• $\log(ab) = \log(a) + \log(b)$—multiplication inside becomes addition outside
• Use this to expand expressions like $\log(5x)$ into $\log(5) + \log(x)$ for easier manipulation
• Both $a$ and $b$ must be positive—logarithms of negative numbers aren't real (this is a common trap in domain questions)

Quotient Rule

• $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$—division inside becomes subtraction outside
• Order matters here: numerator's log comes first, then subtract the denominator's log
• Useful for condensing expressions like $\log(x) - \log(3)$ back into $\log\left(\frac{x}{3}\right)$ when solving equations

Power Rule

• $\log(a^n) = n \cdot \log(a)$—exponents inside become coefficients outside
• Works in reverse too: $3\log(x) = \log(x^3)$, which is essential for condensing expressions
• The exponent $n$ can be any real number—fractions, negatives, whatever (this is how you handle roots: $\log(\sqrt{x}) = \frac{1}{2}\log(x)$)

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Properties of Special Values

These properties emerge directly from the definition and give you instant answers for specific inputs. Memorize these—they're quick points on any assessment.

Logarithm of 1

• $\log(1) = 0$ for any base—because any number raised to the zero power equals 1
• Appears constantly in simplification—whenever you see $\log(1)$, replace it with 0 immediately
• Reflects the exponential identity $10^0 = 1$, reinforcing the logarithm-exponent connection

Logarithm of the Base

• $\log_{a}(a) = 1$ for any valid base $a$—because $a^1 = a$
• For common logs specifically: $\log(10) = 1$, since $10^1 = 10$
• Quick checkpoint for calculations—if your answer involves $\log(10)$, simplify it to 1

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The Inverse Relationship

Logarithms and exponentials undo each other. This inverse relationship is the conceptual heart of solving logarithmic and exponential equations.

Inverse Property

• $10^{\log(x)} = x$ and $\log(10^x) = x$—exponentials and logarithms cancel when their bases match
• This is how you "free" a variable trapped inside a logarithm or exponent during equation solving
• Generalizes to any base: $a^{\log_a(x)} = x$, making it your go-to move for isolating variables

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Converting Between Bases

When you need to evaluate a logarithm your calculator can't handle directly, this formula bridges the gap. It works because all logarithms are proportionally related.

Change of Base Formula

• $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$—converts any logarithm to a ratio of logarithms in your preferred base
• Most common use: $\log_5(20) = \frac{\log(20)}{\log(5)}$, letting you evaluate with a standard calculator
• Works with natural logs too: $\log_a(x) = \frac{\ln(x)}{\ln(a)}$—choose whichever base is more convenient

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Quick Reference Table

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Self-Check Questions

1. Which two rules would you combine to simplify $\log\left(\frac{x^3}{y^2}\right)$ into an expression with no fractions or exponents inside the logarithm?

2. If $\log(a) + \log(b) = \log(c)$, what must be true about the relationship between $a$, $b$, and $c$?

3. Compare and contrast the Product Rule and Power Rule: how does each one change what's inside versus outside the logarithm?

4. You need to solve $5^x = 20$ but only have a common log button. Write out the exact steps using the change of base concept.

5. A student claims that $\log(a - b) = \log(a) - \log(b)$. Explain why this is wrong and identify which rule they confused it with.