# Change of Base in Logarithms Formulas with Examples

 

The proof of the change of base formula for the logarithms is presented

## Change of Base Formula

Let $y = a^x$ and convert it into logarithm to write
$\log_a \; y = x \qquad (1)$

Take the $\log_b$, where $b \gt 0$ and $b \ne 0$ , of both sides of $y = a^x$ to obtain
$\log_b \; y = \log_b \; (a^x)$

Use the rules of logarithms to write $\log_b \; a^x$ as $x \; \log_b \; a$ and substitute in the above equation.
$\log_b \; y = x \log_b \; a$

Substitute $x$ in the above by $\log_a \; y$ from $(1)$ to write $\log_b \; y = \log_a \; y \log_b \; a$

Solve the above for $\log_a \; y$ to obtain the change of base formula $\log_a \; y = \dfrac{\log_b \; y }{\log_b \; a }$ You can chose any base $b$, such that $b \gt 0$ and $b \ne 1$, to rewrite any logarithm.

## Examples of Applications of the Change of Base Formula

### Example 1

a) Evaluate $\log_4 \; 16$ noting that $16 = 4^2$
b) Use the change of base formula to rewrite $\log_4 \; 16$ using $\log$ with base base $2$ and evaluate it again. Compare

Solution
a) $\log_4 \; 16 = \log_4 \; 4^2 = 2$

b) rewrite $\log_4 \; 16$ using the change of base formula with logarithm of base $2$,
$\log_4 \; 16 = \dfrac{\log_2 \; 16 }{\log_2 \; 4 }$

Note that $4 = 2^2$ and $16 = 2^4$. hence
$\log_4 \; 16 = \dfrac{\log_2 \; 2^4 }{\log_2 \; 2^2 } = \dfrac{4}{2} = 2$

The evaluation of $\log_4 \; 16$ in a) and b) gives the same answer as expected.

### Example 2

Express $\log_2 x$ using the natural logarithm $\ln$

Solution
Using the change of base formula with the natural logarithm $\ln$, we write
$\log_2 x = \dfrac{\ln x}{\ln 2}$

The results in example 2 have important implications in calculating derivative and integral of logarithms to any base.