# How do you solve this logarithmic equation? 3log_(5)x-log_(5)(5x)=3-log_(5)25

Jan 8, 2017

$\left\{5\right\}$

#### Explanation:

Put all logarithms to one side:

$3 {\log}_{5} x - {\log}_{5} \left(5 x\right) + {\log}_{5} 25 = 3$

${\log}_{5} 25$ can be rewritten as $\frac{\log 25}{\log 5} = \frac{2 \log 5}{\log 5} = 2$

${\log}_{5} {x}^{3} - {\log}_{5} \left(5 x\right) + 2 = 3$

${\log}_{5} {x}^{3} - {\log}_{5} 5 x = 1$

${\log}_{5} \left(\frac{{x}^{3}}{5 x}\right) = 1$

${x}^{2} / 5 = {5}^{1}$

${x}^{2} = 25$

$x = \pm 5$

The $- 5$ solution is extraneous, since ${\log}_{5} \left(a x\right)$, where $a$ is a positive constant is only defined in the positive-x-values.

Hopefully this helps!