What is 2(log_{a}(z)-3log_{a}(x))+3log_{a}(w) ?

The solution to 2(log_{a}(z)-3log_{a}(x))+3log_{a}(w) is log_{a}((z^2w^3)/(x^6))

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simplify 2(log_{a}(z)-3log_{a}(x))+3log_{a}(w)

simplify

2 (lo g_{a} (z) - 3 lo g_{a} (x)) + 3 lo g_{a} (w)

lo g_{a} (\frac{z ^{2} w ^{3}}{x ^{6}})

Solution steps

2 (lo g_{a} (z) - 3 lo g_{a} (x)) + 3 lo g_{a} (w)

Apply log rule:

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

3 lo g_{a} (x) = lo g_{a} (x^{3})

= 2 (lo g_{a} (z) - lo g_{a} (x^{3})) + 3 lo g_{a} (w)

Apply log rule:

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

lo g_{a} (z) - lo g_{a} (x^{3}) = lo g_{a} (\frac{z}{x ^{3}})

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

Apply log rule:

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

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

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

Apply log rule:

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

3 lo g_{a} (w) = lo g_{a} (w^{3})

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

Apply log rule:

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

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

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

Simplify

(\frac{z}{x ^{3}})^{2} w^{3} : \frac{z ^{2} w ^{3}}{x ^{6}}

= lo g_{a} (\frac{z ^{2} w ^{3}}{x ^{6}})

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

lo g_{6} (\frac{216 x ^{3}}{y})

\frac{1}{20} ln (\frac{31}{20})

ln (2.8246 x 1 0^{- 10})

ln (- iπ)

What is 2(log_{a}(z)-3log_{a}(x))+3log_{a}(w) ?
The solution to 2(log_{a}(z)-3log_{a}(x))+3log_{a}(w) is log_{a}((z^2w^3)/(x^6))