# How do you solve log_5 x^3=15?

Apr 9, 2016

$x = {5}^{5}$

#### Explanation:

${\log}_{5} {x}^{3} = 15$ means ${5}^{15} = {x}^{3}$

As ${\left({5}^{5}\right)}^{3} = {5}^{15}$

${x}^{3} = {\left({5}^{5}\right)}^{3}$ i.e. ${x}^{3} - {\left({5}^{5}\right)}^{3} = 0$, which can be factorized as

$\left(x - {5}^{5}\right) \left({x}^{2} + {5}^{5} x + {\left({5}^{5}\right)}^{2}\right) = 0$

i.e. $x = {5}^{5}$, if we consider the domain only as real numbers.

as ${x}^{2} + {5}^{5} x + {\left({5}^{5}\right)}^{2}$ is of the type ${x}^{2} + a x + {a}^{2}$ and has complex roots because discriminant, which is ${a}^{2} - 4 {a}^{2} = - 3 {a}^{2}$, is always negative.