What Is a Logarithm?

A logarithm answers the question: "What exponent gives me this number?"

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

Converting between exponential and logarithmic forms

Examples:
- $\log_2(8) = 3$ because $2^3 = 8$
- $\log_{10}(1000) = 3$ because $10^3 = 1000$
- $\log_5(25) = 2$ because $5^2 = 25$

$log₂(8) = ?$
$2ˣ = 8$ rewrite as exponential
$2³ = 8$ since 2 $^{3}$ = 8
$log₂(8) = 3$

log₂(8) = 3 because 2³ = 8

Special cases:
- $\log_b(1) = 0$ (because $b^0 = 1$ )
- $\log_b(b) = 1$ (because $b^1 = b$ )
- $\ln(x) = \log_e(x)$ (natural log, $e \approx 2.718$ )

Example 1

Evaluate $\log_3(81)$

Ask: $3^? = 81$

$3^4 = 81$

So $\log_3(81) = 4$

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Theory

Logarithm Properties

Product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$

Quotient rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$

Power rule: $\log_b(x^n) = n \cdot \log_b(x)$

Change of base: $\log_b(x) = \frac{\log(x)}{\log(b)} = \frac{\ln(x)}{\ln(b)}$

Example 1

Simplify $\log_2(8) + \log_2(4)$

Product rule: $\log_2(8 \times 4) = \log_2(32)$

$2^5 = 32$ , so $\log_2(32) = 5$

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