# How do you simplify log_20 (8000^x)?

Sep 11, 2016

$3 x$

#### Explanation:

Recall:

The " power law " of logs states:

${\log}_{p} {q}^{\textcolor{red}{m}} = \textcolor{red}{m} {\log}_{p} q$

The " change of base law " states

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

($\text{We usually use " log_10" as } {\log}_{c}$)

Apply these two laws to the question:

${\log}_{20} {8000}^{\textcolor{red}{x}} = \textcolor{red}{x} {\log}_{20} 8000$

=$\textcolor{red}{x} \left(\frac{{\log}_{10} 8000}{{\log}_{10} 20}\right) \text{ } \leftarrow$ use a calculator

=$x \times 3 = 3 x$

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However, using another approach - the maths is beautiful, and we do not even need a calculator this time!

Note that ${20}^{3} = 8000$

${\log}_{20} {8000}^{x} = {\log}_{20} {\left({20}^{3}\right)}^{x}$

=${\log}_{20} {20}^{3 x}$

=$3 x {\log}_{20} 20 \text{ } \leftarrow {\log}_{20} 20 = 1$

$3 x$