How do you solve (¼)log_6 (a – 3) – log_6 3 = 0?

Jun 5, 2016

I found: $a = 84$

Explanation:

We can first get rid of $\frac{1}{4}$:
${\log}_{6} {\left(a - 3\right)}^{\frac{1}{4}} - {\log}_{6} \left(3\right) = 0$
Then we change the subtraction operating on the arguments:
${\log}_{6} \left[{\left(a - 3\right)}^{\frac{1}{4}} / 3\right] = 0$
We use the definition of log:
${\left(a - 3\right)}^{\frac{1}{4}} / 3 = {6}^{0}$
${\left(a - 3\right)}^{\frac{1}{4}} / 3 = 1$
rearrange:
${\left(a - 3\right)}^{\frac{1}{4}} = 3$
take the power of $4$ on both sides:
${\left(a - 3\right)}^{\frac{1}{4} \cdot 4} = {3}^{4}$
$a - 3 = 81$
$a = 84$