How do you calculate ${log}_{\frac{1}{625}} 125$ $log_(1/625) 125$ ?

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Answer 1 · 2016-04-22T11:05:59

${log}_{\frac{1}{625}} (125) = - \frac{3}{4}$ $log_(1/625)(125)= -3/4$

We will use the following:

$y = {log}_{a} (x) \Leftrightarrow a^{y} = x$ $y = log_a(x) <=> a^y = x$ (for $x > 0$ $x>0$ )
${log}_{a} (a^{x}) = x$ $log_a(a^x) = x$
${(a^{b})}^{c} = a^{b c}$ $(a^b)^c = a^(bc)$
$a^{- b} = \frac{1}{a^{b}}$ $a^(-b) = 1/a^b$

Let $x = {log}_{\frac{1}{625}} (125)$ $x = log_(1/625)(125)$

$\Rightarrow {(\frac{1}{625})}^{x} = 125$ $=> (1/625)^x = 125$

$\Rightarrow {(5^{- 4})}^{x} = 5^{3}$ $=> (5^(-4))^x = 5^3$

$\Rightarrow 5^{- 4 x} = 5^{3}$ $=> 5^(-4x) = 5^3$

$\Rightarrow {log}_{5} (5^{- 4 x}) = {log}_{5} (5^{3})$ $=> log_5(5^(-4x)) = log_5(5^3)$

$\Rightarrow - 4 x = 3$ $=> -4x = 3$

$:. x = -3/4$

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