The calculators which we use in our day today life have only two types of logarithm keys, one is common log (base 10) and one for natural log (base e). By using the change of base formula to express logarithms in terms of common logarithms or natural logarithms, one can evaluate a logarithms to any base. In this section we will be learning more about change of base formula.

For some purpose, we find it useful to change from logarithms in one base to logarithm in another base. Let $a, b$ and x be positive real numbers such that a $\neq$ 1 and b $\neq 1$.

And suppose we are given $\log_{a}x$ and want to find $\log_{b}x$. Let

$y$ = $\log_{b} x$

we can write this in exponential form given as

$b^{y} = x$

Take the logarithm, with base $a$, of each side

$\log_{a} b^{y}$ = $\log_{a} x$

$y log_{a} b$ = $\log_{a} x$

Divide the equation by $\log_{a}b$

$y$ = $\frac{\log_{a} x}{\log_{a} b}$
The Change of Base Formula

$\log _{b} x$ = $\frac{\log_{a} x}{\log_{a} b}$
It can also be written as

$\log_{b}x$ = $\frac{1}{\log_{a} b}$$\log_{a} x$

So the $\log_{b} x$ is just a constant multiple of $\log_{a} x$:

The constant is $\frac{1}{\log_{a} b}$

If $x = a$, then $\log_{a} a = 1$ and the above formula becomes

$\log_{b} a$ = $\frac{1}{\log_{a} b}$

1. The change of base formula for converting any logarithms to common logarithms (base 10)

$\log_{b}x$ = $\frac{\log_{10} x}{\log_{10} b}$

2. The change of base formula for converting any logarithms to natural logarithms (base e)

$\log_{b}x$ = $\frac{\log_{e} x}{\log_{e} b}$

Solved Examples

Question 1: Use change of base formula and evaluate the following logarithm. 

$\log_{6} 3$

Solution:
 
Change of base formula  $\log_{b}x$ = $\frac{\log_{a} x}{\log_{a} b}$

Given, $b = 6$ and $a = 10$ (common logarithms)

$\log_{6} 3$ = $\frac{\log_{10} 3}{\log_{10} 6}$ = $0.6131472$

 

Question 2:
Use change of base formula and evaluate the following logarithm. 

$\log_{5} 8$

Solution:
 
Change of base formula  $\log_{b}x$ = $\frac{\log_{a} x}{\log_{a} b}$

Given, $b = 5$ and $a = e$ (natural logarithms)

$\log_{5} 8$ = $\frac{\log_{e} 8}{\log_{e} 5}$ = $1.2920296$
 

Question 3: Solve $\log_{18}60$.
Solution:
 
Change of base formula  $\log_{b}x$ = $\frac{\log_{a} x}{\log_{a} b}$

Given, $b = 18$ and $a = 10$ (common logarithms)

$\log_{6} 3$ = $\frac{\log_{10} 60}{\log_{10} 18}$ = $1.4165460$