# How do you solve logₐx^10-2logₐ(x^3/4)=4logₐ(2x)?

Nov 27, 2015

The equation is an identity, and therefore it has an infinite number of solutions.

#### Explanation:

${\log}_{a} \left({x}^{10}\right) - 2 {\log}_{a} \left({x}^{3} / 4\right) = 4 {\log}_{a} \left(2 x\right)$
let's work the left side$\left(L S\right)$ only and simplify it:
$L S = {\log}_{a} \left({x}^{10}\right) - {\log}_{a} \left({x}^{6} / 16\right)$
$L S = {\log}_{a} \left\{\frac{{x}^{10}}{\frac{{x}^{6}}{16}}\right\}$
$L S = {\log}_{a} \left(16 {x}^{4}\right)$
$L S = {\log}_{a} \left[{\left(2 x\right)}^{4}\right]$
$L S = 4 {\log}_{a} \left(2 x\right)$
Now:
$R S = 4 {\log}_{a} \left(2 x\right)$
$\therefore L S = R S$
Within the basic laws and valid domain of logarithms, i.e: $a \ne 1 \mathmr{and} x > 0$, the original equation is an identity, and therefore it has an infinite number of solutions.