Change-of-base formula

Definition

The change-of-base formula is used to rewrite logarithms in terms of logs of another base, allowing for easier computation. It is commonly written as $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ where $b$ and $c$ are positive real numbers and $c \neq 1$.

5 Must Know Facts For Your Next Test

1. The change-of-base formula helps to evaluate logarithms with bases other than 10 or e using a calculator.
2. It can be applied to any logarithmic base conversion, not just between common log (base-10) and natural log (base-e).
3. A common application is converting $\log_b(a)$ to either $\log_{10}(a)$ or $\ln(a)$ for ease of calculation.
4. Understanding this formula is essential for solving complex logarithmic equations that involve different bases.
5. It illustrates the fundamental property that logarithmic functions of different bases are proportional to each other.

Review Questions

• How does the change-of-base formula simplify the computation of $\log_2(8)$?
• Convert $\log_3(9)$ using the change-of-base formula with base-10.
• Why is the change-of-base formula useful when using calculators?

Related terms

Logarithm: A mathematical function that determines how many times one number must be multiplied by itself to obtain another number. Represented as $\log_b(a)$ where $b$ is the base.

$\ln(x)$: $\ln(x)$ denotes the natural logarithm, which has a base of e (approximately equal to 2.718).

$\log_{10}(x)$: $\log_{10}(x)$ denotes the common logarithm, which has a base of 10.