# How do you calculate log_7 5.8 with a calculator?

Aug 9, 2016

${\log}_{7} 5.8 = \frac{{\log}_{10} 5.8}{{\log}_{10} 7} = 0.9034$

#### Explanation:

Some calculators are able to calculate logs with any base, but let's work through this for those calculators which can only work with base 10.

Let ${\log}_{7} 5.8 = x \text{ log form}$

${7}^{x} = 5.8 \text{ index form}$

The variable is in the index, find the log of both sides.

${\log}_{10} {7}^{x} = {\log}_{10} 5.8$

$x {\log}_{10} 7 = {\log}_{10} 5.8$

$x = \frac{{\log}_{10} 5.8}{{\log}_{10} 7}$

Now use a calculator to find the answer $0.9034$

${\log}_{a} b = \frac{{\log}_{10} b}{{\log}_{10} a} \text{ is called the change of base law}$

We could have written the following in one step:

${\log}_{7} 5.8 = \frac{{\log}_{10} 5.8}{{\log}_{10} 7}$