 # Change of Base Formula

The Change of base formula helps to rewrite the logarithm in terms of another base log. Change of base formula is used in the evaluation of log and have another base than 10.

$\LARGE \log_{b}x=\frac{\log _{d}x}{\log _{d}b}$

### Solved Examples

Example 1:
Solve: 2
$$\begin{array}{l}\log_{4} 29\end{array}$$

Solution:

Given,
2
$$\begin{array}{l}\log_{4} 29\end{array}$$

Using logarithm change of base formula,

$$\begin{array}{l}log_{b} x\end{array}$$
=
$$\begin{array}{l}\frac{log_{d} x}{log_{d} b}\end{array}$$

2

$$\begin{array}{l}\log_{4} 29\end{array}$$
= 2
$$\begin{array}{l}\times\end{array}$$

$$\begin{array}{l}\frac{log_{10} 29}{log_{10} 4}\end{array}$$

2
$$\begin{array}{l}\log_{4} 29\end{array}$$
= 2 x 2.43 = 4.86

Example 2: Simply:

$$\begin{array}{l}\log_{32} 16\end{array}$$

Solution:

Given,

$$\begin{array}{l}\log_{32} 16\end{array}$$

Using change of base formula,

$$\begin{array}{l}\log_{32} 16=\frac{\log_{10}16}{\log_{10}32}\\=\frac{\log_{10}2^4}{\log_{10}2^5}\\=\frac{4\log_{10}2}{5\log_{10}2}\\=\frac{4}{5}\end{array}$$