What is the change of base formula?

The change of base formula, shown below, is used to change the base of a logarithm from one value to another.

$$\begin{align*} \log_a{x}=\frac{\log_b{x}}{\log_b{a}} \end{align*}$$

Where $a,b \neq 1$ and $x>0$.

We can transform a logarithm with base $a$ into the expression on the R.H.S. with base $b$. The formula is particularly useful as we can choose any value we like for $b$.

We can use the formula for a range of different things, such as

• simplifying expressions with logarithms
• solving equations with logarithms
• comparing the size/value of logarithms with different bases

For example, if we want to simplify $\log_2{x}+\log_4{y}$ we need to change one of the terms in the expression so that they both have the same base. This then allows us to use other logarithm laws.

Let's use the formula to change the logarithm in the second term to base $2$.

So that means we choose $b=2$. Using other well-known logarithm laws, we get.

$$\begin{align*} \log_2{x}+\log_4{y}&=\log_2{x}+\frac{\log_2{y}}{\log_2{4}}\hspace{2em}(\text{recall }\rightarrow 2^2=4)\\[12pt] &=\log_2{x}+\frac{\log_2{y}}{2}\\[12pt] &=\log_2{x}+\log_2{\sqrt{y}}\\[12pt] &=\log_2{x\sqrt{y}} \end{align*}$$