What is log_{sqrt(125)}(625) ?

The solution to log_{sqrt(125)}(625) is 8/3

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simplify log_{sqrt(125)}(625)

simplify

lo g_{125} (625)

\frac{8}{3}

Decimal Notation

2.66666 \dots

Solution steps

lo g_{125} (625)

Factor

125 : 5^{\frac{3}{2}}

= lo g_{5^{\frac{3}{2}}} (625)

Apply log rule:

lo g_{a^{b}} (x) = \frac{1}{b} lo g_{a} (x), a > 0

lo g_{5^{\frac{3}{2}}} (625) = \frac{1}{\frac{3}{2}} lo g_{5} (625)

= \frac{1}{\frac{3}{2}} lo g_{5} (625)

Factor the number:

625 = 5^{4}

= \frac{1}{\frac{3}{2}} lo g_{5} (5^{4})

Apply log rule:

lo g_{a} (a^{b}) = b

lo g_{5} (5^{4}) = 4

= \frac{1}{\frac{3}{2}} \cdot 4

\frac{1}{\frac{3}{2}} \cdot 4 = \frac{8}{3}

= \frac{8}{3}

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