# How do you solve log _x 8 = -3?

Dec 4, 2015

$x = \frac{1}{2}$

#### Explanation:

${\log}_{x} 8 = - 3$
$\implies {\log}_{x} {2}^{3} = - 3$
$3 {\log}_{x} 2 = - 3$ (The logarithm of the ${3}^{r d}$ power of a number is $3$ times the logarithm of the number itself)
Divide both side into $\frac{1}{3}$, in order to simplify left side:
$\frac{1}{3} \cdot 3 {\log}_{x} 2 = \frac{1}{3} \cdot \left(- 3\right)$
$\implies {\log}_{x} 2 = - 1$
$\implies {x}^{-} 1 = 2$ (definition of logarithm)
Multiply both side by $x$:
$x \cdot {x}^{-} 1 = x \cdot 2$
$\implies 2 x = {x}^{0}$ (If we take the product of two exponentials with the same nonzero base, we simply add the exponents)
Multiply both side by $\frac{1}{2}$, in order to simplify left side:
$\frac{1}{2} \cdot 2 x = \frac{1}{2} \cdot 1$
$x = \frac{1}{2}$