What is log_{5}(4)+1/3 log_{5}(x) ?

The solution to log_{5}(4)+1/3 log_{5}(x) is log_{5}(4\sqrt[3]{x})

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simplify log_{5}(4)+1/3 log_{5}(x)

simplify

lo g_{5} (4) + \frac{1}{3} lo g_{5} (x)

Solution steps

lo g_{5} (4) + \frac{1}{3} lo g_{5} (x)

lo g_{5} (4) = 2 lo g_{5} (2)

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

2 lo g_{5} (2) = lo g_{5} (2^{2})

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

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

Apply log rule:

lo g_{a} (x) + lo g_{a} (y) = lo g_{a} (x y)

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

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

Simplify

2^{2} x^{\frac{1}{3}} : 4 x^{\frac{1}{3}}

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

Apply exponent rule:

ln (\frac{6}{5})

\frac{15}{2} ln (\frac{2}{5})

\frac{3}{2} [ln (x^{2} - 1) - ln (x + 1) + ln (x - 1)]

ln (x \frac{y}{z})

lo g_{10} ((\frac{6}{5})^{6})

What is log_{5}(4)+1/3 log_{5}(x) ?
The solution to log_{5}(4)+1/3 log_{5}(x) is log_{5}(4\sqrt[3]{x})