Logarithm Properties

Why This Matters

Logarithm properties aren't just formulas to memorize—they're the key to unlocking exponential equations that would otherwise be impossible to solve. When you're tested on logarithms, you're really being tested on your ability to recognize when and how to apply these properties to simplify expressions, solve equations, and convert between forms. These properties show up everywhere: in exponential growth and decay problems, in calculus when you're differentiating logarithmic functions, and in real-world applications like measuring earthquake intensity or sound decibels.

Here's the thing: every logarithm property exists because logarithms are the inverse of exponentiation. That single relationship explains why the product rule turns multiplication into addition, why the power rule pulls exponents down, and why $a^{\log_a(x)} = x$. Don't just memorize these rules in isolation—understand that they're all connected through this inverse relationship. When you see a complex logarithmic expression on an exam, your job is to identify which property transforms it into something solvable.

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Expanding and Condensing Expressions

These three properties form the core toolkit for manipulating logarithmic expressions. They convert operations inside the logarithm (multiplication, division, powers) into simpler operations outside (addition, subtraction, scalar multiplication).

Product Rule

• $\log_a(xy) = \log_a(x) + \log_a(y)$—multiplication inside becomes addition outside
• Expanding expressions with this rule is essential when you need to isolate variables that are multiplied together
• Derived from exponent rules: since $a^m \cdot a^n = a^{m+n}$, the inverse operation splits products into sums

Quotient Rule

• $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$—division inside becomes subtraction outside
• Order matters: the numerator's log comes first, then subtract the denominator's log
• Pairs with product rule when condensing expressions—watch for addition (product) vs. subtraction (quotient)

Power Rule

• $\log_a(x^n) = n \cdot \log_a(x)$—exponents drop down as coefficients
• Most powerful for solving equations where the variable is in an exponent (take the log of both sides, then apply this rule)
• Works with any exponent: fractions ($\log_a(\sqrt{x}) = \frac{1}{2}\log_a(x)$), negatives, and variables

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Converting Between Bases

When your calculator only has $\log$ (base 10) and $\ln$ (base e), this property saves the day.

Change of Base Formula

• $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$—convert any logarithm to a more convenient base
• Calculator application: $\log_5(20) = \frac{\log(20)}{\log(5)} = \frac{\ln(20)}{\ln(5)}$—both give the same answer
• Useful for comparing logarithms with different bases or graphing logarithmic functions with non-standard bases

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Fundamental Identity Properties

These properties establish the basic behavior of logarithms and are essential reference points for simplification. They follow directly from the definition: $\log_a(x) = y$ means $a^y = x$.

Logarithm of 1

• $\log_a(1) = 0$ for any valid base $a$—because $a^0 = 1$ for all positive $a \neq 1$
• Simplification shortcut: any term with $\log(1)$ vanishes from your expression
• Graphical meaning: every logarithmic function passes through the point $(1, 0)$

Logarithm of the Base

• $\log_a(a) = 1$ always—because $a^1 = a$
• Quick simplification: $\log_2(2) = 1$, $\ln(e) = 1$, $\log(10) = 1$
• Extends to powers of the base: $\log_a(a^n) = n$ by combining with the power rule

Inverse Property

• $a^{\log_a(x)} = x$ and $\log_a(a^x) = x$—logarithms and exponentials undo each other
• The defining relationship that makes logarithms useful for solving exponential equations
• Application: to solve $3^x = 17$, take $\log_3$ of both sides: $x = \log_3(17)$

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Solving Equations with Logarithms

These properties establish when equations have solutions and how to find them.

Equality Property

• If $\log_a(x) = \log_a(y)$, then $x = y$—same base, same output means same input
• Primary solving technique: condense both sides to a single logarithm, then drop the logs and solve
• Works in reverse: if $x = y$, then $\log_a(x) = \log_a(y)$—useful for taking logs of both sides

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Domain Restrictions

Understanding where logarithms exist prevents errors and helps you check solutions.

Domain of Logarithms

• The argument must be positive: $x > 0$ for $\log_a(x)$—no zero, no negatives allowed inside
• Extraneous solutions often violate this rule—always check that your answer makes the argument positive
• Base restrictions: $a > 0$ and $a \neq 1$ (base must be positive and not equal to 1)

Logarithm of Non-Positive Numbers

• $\log_a(x)$ is undefined for $x \leq 0$ in real numbers—there's no real power that makes a positive base negative or zero
• Critical for checking answers: if solving $\log(x-3) = 2$ gives $x = 1$, reject it since $1-3 < 0$
• Complex logarithms exist but are beyond the scope of algebra and trigonometry courses

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Quick Reference Table

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Self-Check Questions

1. Which two properties would you use together to expand $\log_3\left(\frac{x^2y}{z}\right)$ completely?

2. You solve a logarithmic equation and get $x = -5$ and $x = 2$. The original equation contained $\log(x+3)$. Which solution(s) are valid, and why?

3. Compare and contrast the Product Rule and Power Rule: how does each handle $\log(x \cdot x \cdot x)$, and do they give the same result?

4. Using the Change of Base Formula, explain why $\frac{\log_2(8)}{\log_2(4)}$ is not equal to $\log_4(8)$—what common mistake does this test?

5. If an equation simplifies to $\log_5(2x-1) = \log_5(x+4)$, what property allows you to solve it, and what must you verify about your answer?