What is 1/3 log_{m}(y)+6log_{m}(w)-log_{m}(z) ?

The solution to 1/3 log_{m}(y)+6log_{m}(w)-log_{m}(z) is log_{m}((\sqrt[3]{y}w^6)/z)

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simplify 1/3 log_{m}(y)+6log_{m}(w)-log_{m}(z)

simplify

\frac{1}{3} lo g_{m} (y) + 6 lo g_{m} (w) - lo g_{m} (z)

Solution steps

\frac{1}{3} lo g_{m} (y) + 6 lo g_{m} (w) - lo g_{m} (z)

\frac{1}{3} lo g_{m} (y) = lo g_{m} (y^{\frac{1}{3}})

6 lo g_{m} (w) = lo g_{m} (w^{6})

= lo g_{m} (y^{\frac{1}{3}}) + lo g_{m} (w^{6}) - lo g_{m} (z)

Apply log rule:

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

lo g_{m} (y^{\frac{1}{3}}) + lo g_{m} (w^{6}) = lo g_{m} (y^{\frac{1}{3}} w^{6})

= lo g_{m} (y^{\frac{1}{3}} w^{6}) - lo g_{m} (z)

Apply log rule:

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

lo g_{m} (y^{\frac{1}{3}} w^{6}) - lo g_{m} (z) = lo g_{m} (\frac{y ^{\frac{1}{3}} w ^{6}}{z})

= lo g_{m} (\frac{y ^{\frac{1}{3}} w ^{6}}{z})

Apply exponent rule:

ln (2 ln (2)) - ln (ln (2))

lo g_{6} (36 x^{6})

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

ln (\frac{x ^{4}}{( 2 x - 1 ) ^{3} ( 7 x - 5 ) ^{8}})

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

What is 1/3 log_{m}(y)+6log_{m}(w)-log_{m}(z) ?
The solution to 1/3 log_{m}(y)+6log_{m}(w)-log_{m}(z) is log_{m}((\sqrt[3]{y}w^6)/z)