What is 2log_{m}(w)-1/3 log_{m}(x)+3log_{m}(z) ?

The solution to 2log_{m}(w)-1/3 log_{m}(x)+3log_{m}(z) is log_{m}((w^2z^3x^{2/3})/x)

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simplify 2log_{m}(w)-1/3 log_{m}(x)+3log_{m}(z)

simplify

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

lo g_{m} (\frac{w ^{2} z ^{3} x ^{\frac{2}{3}}}{x})

Solution steps

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

Apply log rule:

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

2 lo g_{m} (w) = lo g_{m} (w^{2})

= lo g_{m} (w^{2}) - \frac{1}{3} lo g_{m} (x) + 3 lo g_{m} (z)

Apply log rule:

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

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

= lo g_{m} (w^{2}) - lo g_{m} (x^{\frac{1}{3}}) + 3 lo g_{m} (z)

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

lo g_{m} (w^{2}) - lo g_{m} (x^{\frac{1}{3}}) = lo g_{m} (\frac{w ^{2}}{x ^{\frac{1}{3}}})

= lo g_{m} (\frac{w ^{2}}{x ^{\frac{1}{3}}}) + 3 lo g_{m} (z)

Simplify

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

= lo g_{m} (\frac{w ^{2} x ^{\frac{2}{3}}}{x}) + 3 lo g_{m} (z)

Apply log rule:

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

3 lo g_{m} (z) = lo g_{m} (z^{3})

= lo g_{m} (\frac{w ^{2} x ^{\frac{2}{3}}}{x}) + lo g_{m} (z^{3})

Apply log rule:

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

lo g_{m} (\frac{w ^{2} x ^{\frac{2}{3}}}{x}) + lo g_{m} (z^{3}) = lo g_{m} (\frac{w ^{2} x ^{\frac{2}{3}}}{x} z^{3})

= lo g_{m} (\frac{w ^{2} x ^{\frac{2}{3}}}{x} z^{3})

Simplify

\frac{w ^{2} x ^{\frac{2}{3}}}{x} z^{3} : \frac{w ^{2} z ^{3} x ^{\frac{2}{3}}}{x}

= lo g_{m} (\frac{w ^{2} z ^{3} x ^{\frac{2}{3}}}{x})

lo g_{10} (4) + 2 lo g_{10} (x)

ln (\frac{x + 2}{e ^{x}})

ln (\frac{1}{9 + 9 e ^{x}} \cdot \frac{1}{8 + 8 e ^{x}})

What is 2log_{m}(w)-1/3 log_{m}(x)+3log_{m}(z) ?
The solution to 2log_{m}(w)-1/3 log_{m}(x)+3log_{m}(z) is log_{m}((w^2z^3x^{2/3})/x)