# How do you solve log_6 x -log_6(x-6)=1?

Jan 9, 2017

$x = 7.2$

#### Explanation:

Using the Rule$: {\log}_{m} a - {\log}_{m} b = {\log}_{m} \left(\frac{a}{b}\right)$, we get,

${\log}_{6} \left(\frac{x}{x - 6}\right) = 1. \ldots \ldots \ldots \ldots . . \left(1\right)$

Since, ${\log}_{m} a = n \Rightarrow {m}^{n} = a$, we have, from $\left(1\right)$,

$\frac{x}{x - 6} = {6}^{1} = 6$

$\therefore x = 6 \left(x - 6\right) = 6 x - 36$

$\therefore 5 x = 36 \Rightarrow x = \frac{36}{5} = 7.2$

This root satisfy the given eqn.

Hence, $x = 7.2$ is the Soln.

Jan 9, 2017

Use $\log \left(a\right) - \log \left(b\right) = \log \left(\frac{a}{b}\right)$ to combine to a single log, eliminate the log by making both sides a power of the base, then solve for x.

#### Explanation:

Use the property $\log \left(a\right) - \log \left(b\right) = \log \left(\frac{a}{b}\right)$:

${\log}_{6} \left(\frac{x}{x - 6}\right) = 1$

Make the log disappear by making both sides the exponent of 6:

$\frac{x}{x - 6} = {6}^{1}$

Solve for x:

$x = 6 x - 36$

$- 5 x = - 36$

$x = \frac{36}{5}$

Check:

${\log}_{6} \left(\frac{36}{5}\right) - {\log}_{6} \left(\frac{36}{5} - 6\right) = 1$

${\log}_{6} \left(36\right) - {\log}_{6} \left(5\right) - {\log}_{6} \left(6\right) + {\log}_{5} \left(5\right) = 1$

$2 {\log}_{6} \left(6\right) - {\log}_{6} \left(5\right) - {\log}_{6} \left(6\right) + {\log}_{5} \left(5\right) = 1$

${\log}_{6} \left(6\right) = 1$

$1 = 1$

This checks.