What is log_{5}(a)+1/3 log_{5}(b) ?

The solution to log_{5}(a)+1/3 log_{5}(b) is log_{5}(ab^{1/3})

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simplify log_{5}(a)+1/3 log_{5}(b)

simplify

lo g_{5} (a) + \frac{1}{3} lo g_{5} (b)

lo g_{5} (a b^{\frac{1}{3}})

Solution steps

lo g_{5} (a) + \frac{1}{3} lo g_{5} (b)

Apply log rule:

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

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

= lo g_{5} (a) + lo g_{5} (b^{\frac{1}{3}})

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

lo g_{5} (a) + lo g_{5} (b^{\frac{1}{3}}) = lo g_{5} (a b^{\frac{1}{3}})

= lo g_{5} (a b^{\frac{1}{3}})

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What is log_{5}(a)+1/3 log_{5}(b) ?
The solution to log_{5}(a)+1/3 log_{5}(b) is log_{5}(ab^{1/3})