### Change of Base of Logarithms

${\mathrm{log}}_{a}b=\frac{{\mathrm{log}}_{c}b}{{\mathrm{log}}_{c}a}$

and

${\mathrm{log}}_{a}b=\frac{1}{{\mathrm{log}}_{b}a}$

and

${\mathrm{log}}_{a}b=\frac{1}{{\mathrm{log}}_{b}a}$

**:**

Example

Example

Find the value of the following:

a. ${\mathrm{log}}_{25}100$

b. ${\mathrm{log}}_{3}0.45$

**Answer**:

a.

$\begin{array}{l}{\mathrm{log}}_{25}100\\ =\frac{{\mathrm{log}}_{10}100}{{\mathrm{log}}_{10}25}\\ =\frac{10}{1.3979}\\ =7.154\end{array}$

b.

$\begin{array}{l}{\mathrm{log}}_{3}0.45\\ =\frac{{\mathrm{log}}_{10}0.45}{{\mathrm{log}}_{10}3}\\ =\frac{-0.3468}{0.4771}\\ =-0.727\end{array}$

a. log 100 should be log 10 to the power of 2, equal 2