# How do you find the value of log_12 18 using the change of base formula?

Nov 1, 2016

= ${\log}_{12} 18 = 1.163$

#### Explanation:

Some calculators can work with different bases, but some only have ${\log}_{10}$.
It is useful to understand where the change of base rule comes from.

Log form and index form are interchangeable.

${\log}_{12} 18 = x \Leftrightarrow {12}^{x} = 18 \text{ } \leftarrow$ x must be 1.??????

$\textcolor{w h i t e}{\times \times \times x} {\log}_{10} {12}^{x} = {\log}_{10} 18 \text{ } \leftarrow$ log both sides

$\textcolor{w h i t e}{\times \times \times x} x {\log}_{10} 12 = {\log}_{10} 18 \text{ } \leftarrow$ log power law

$\textcolor{w h i t e}{\times \times \times \times \times \times} x = \frac{{\log}_{10} 18}{{\log}_{10} 12} \text{ } \leftarrow$ solve for x

$\textcolor{w h i t e}{\times \times \times \times \times \times} x = \frac{1.2552725}{1.0791812} \text{ } \leftarrow$ calculate x

$\textcolor{w h i t e}{\times \times \times \times \times \times} x = 1.1631712 \text{ } \leftarrow$ calculate x

Now answer to whatever level of accuracy is required.
Usually 2 or 3 decimal places is sufficient.

This is called the "change of base rule" - the base has been changed from 12 in the given example to base 10 for the numerator and denominator. Now you can calculate the answer using any scientific calculator (or even find the values on a table)

${\log}_{a} b = \frac{{\log}_{c} b}{{\log}_{c} a} \text{ } \leftarrow ' c '$ is usually 10, can be any base