How do you solve ${log}_{x} 8 = - 3$ $log _x 8 = -3$ ?

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How do you solve ${log}_{x} 8 = - 3$ $log _x 8 = -3$ ?

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Answer 1 · 2015-12-04T10:24:16

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

Explanation:

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

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How do you solve ${log}_{x} 8 = - 3$ $log _x 8 = -3$ ?

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