Change-of-base formula

Definition

The change-of-base formula allows you to evaluate logarithms with any base using logarithms of different bases. It is given by the formula $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$, where $c$ is any positive number.

5 Must Know Facts For Your Next Test

1. The change-of-base formula is applicable for any positive numbers $a$, $b$, and $c$, where $b \neq 1$ and $c \neq 1$.
2. You can use common logarithms (base 10) or natural logarithms (base e) as the new base in the change-of-base formula.
3. The change-of-base formula simplifies complex logarithmic expressions that cannot be easily calculated by hand.
4. It is particularly useful for evaluating logarithms on a calculator that may only support base 10 or base e.
5. The change-of-base formula can also be used to derive other logarithmic properties, such as converting between different logarithmic bases.

Review Questions

• What is the change-of-base formula?
• Why might you use common or natural logarithms when applying the change-of-base formula?
• How would you apply the change-of-base formula to evaluate $\log_3(7)$?

Related terms

Logarithm: A mathematical function that determines how many times one number, called the base, must be multiplied by itself to obtain another number.

Natural Logarithm: A logarithm with base $e$, where $e \approx 2.71828$. It is denoted as $\ln(x)$.

Exponential Function: A mathematical function of the form $f(x) = b^x$, where $b$ is a positive constant called the base and $x$ is an exponent.