What is 2log_{a}(5)+log_{a}(6)-1/3 log_{a}(8) ?

The solution to 2log_{a}(5)+log_{a}(6)-1/3 log_{a}(8) is log_{a}(75)

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simplify 2log_{a}(5)+log_{a}(6)-1/3 log_{a}(8)

simplify

2 lo g_{a} (5) + lo g_{a} (6) - \frac{1}{3} lo g_{a} (8)

lo g_{a} (75)

Solution steps

2 lo g_{a} (5) + lo g_{a} (6) - \frac{1}{3} lo g_{a} (8)

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

- \frac{1}{3} lo g_{a} (8) = - lo g_{a} (8^{\frac{1}{3}})

= lo g_{a} (5^{2}) + lo g_{a} (6) - lo g_{a} (8^{\frac{1}{3}})

Apply log rule:

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

lo g_{a} (5^{2}) + lo g_{a} (6) = lo g_{a} (5^{2} \cdot 6)

= lo g_{a} (5^{2} \cdot 6) - lo g_{a} (8^{\frac{1}{3}})

Apply log rule:

lo g_{a} (x) - lo g_{a} (y) = lo g_{a} (\frac{x}{y})

lo g_{a} (5^{2} \cdot 6) - lo g_{a} (8^{\frac{1}{3}}) = lo g_{a} (\frac{5 ^{2} \cdot 6}{8 ^{\frac{1}{3}}})

= lo g_{a} (\frac{5 ^{2} \cdot 6}{8 ^{\frac{1}{3}}})

Cancel

\frac{5 ^{2} \cdot 6}{8 ^{\frac{1}{3}}} : 75

= lo g_{a} (75)

5 + lo g_{10} (3 x)

3 lo g_{2} (x) + 2 lo g_{2} (y) - \frac{1}{2} lo g_{2} (z)

lo g_{7} (\frac{x ^{2} y ^{3}}{z})

ln (\frac{x y}{y ^{3}})

What is 2log_{a}(5)+log_{a}(6)-1/3 log_{a}(8) ?
The solution to 2log_{a}(5)+log_{a}(6)-1/3 log_{a}(8) is log_{a}(75)