# How do you solve 3 log_5 x - log_5 4 = log_5 16?

Jun 18, 2015

The answer is $x = 4$

#### Explanation:

To solve this equation you have to use the facts that:

1. $a \cdot {\log}_{b} \left(c\right) = {\log}_{b} \left({c}^{a}\right)$
2. ${\log}_{a} \left(b\right) - {\log}_{a} \left(c\right) = {\log}_{a} \left(\frac{b}{c}\right)$

First you use (1) to get:

${\log}_{5} \left({x}^{3}\right) - {\log}_{5} \left(4\right) = {\log}_{5} \left(16\right)$

Then you use (2) to get:

${\log}_{5} \left({x}^{3} / 4\right) = {\log}_{5} \left(16\right)$

Now you can leave the logarithms (they both have the same base)

${x}^{3} / 4 = 16$

${x}^{3} = 64$

$x = 4$ (because ${4}^{3} = 4 \cdot 4 \cdot 4 = 64$)