Most common scientific calculators have a key for computing logarithms with base $10$, but do not have keys for other bases. So, if you needed to get an approximation to a number like $\log_4(9)$ it can be difficult to do so. One could easily guess that it is between $1$ and $2$ since $9$ is between $4^1$ and $4^2$, but it is difficult to get an accurate approximation. Fortunately, there is a change of base formula that can help.

Change of Base Formula

The change of base formula for logarithms is:

$\displaystyle \log_a(x)=\frac{\log_b(x)}{\log_b(a)}$

Thus, for example, we could calculate that $\log_4(9)=\frac{\log_{10}(9)}{\log_{10}(4)}$ which could be computed on almost any handheld calculator. 

Deriving the Formula

To see why the formula is true, give $\log_a(x)$ a name like $z$:

$\displaystyle z=\log_a(x)$ 

Write this as $a^z=x$

Now take the logarithm with base $b$ of both sides, yielding:

$\displaystyle \log_b a^z = \log_bx$ 

Using the power rule gives:

 $\displaystyle z \cdot \log_ba = \log_b x$

Dividing both sides by $\log_ba$ gives:

 $\displaystyle z={\log_b x \over \log_ba}.$

Thus we have $\log_a x ={\log_b x \over \log_b a}. $

Example

An expression of the form $\log_5(10^{x^2+1})$ might be easier to graph on a graphing calculator or other device if it were written in base $10$ instead of base 5. The change-of-base formula can be applied to it:

$\displaystyle \log_5(10^{x^2+1}) = {\log_{10}(10^{x^2+1}) \over \log_{10}5}$ 

Which can be written as ${x^2+1 \over \log_{10} 5}. $