What is 2(log_{6}(x)-2log_{6}(z))+3log_{6}(w) ?

The solution to 2(log_{6}(x)-2log_{6}(z))+3log_{6}(w) is log_{6}((x^2w^3)/(z^4))

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simplify 2(log_{6}(x)-2log_{6}(z))+3log_{6}(w)

simplify

2 (lo g_{6} (x) - 2 lo g_{6} (z)) + 3 lo g_{6} (w)

lo g_{6} (\frac{x ^{2} w ^{3}}{z ^{4}})

Solution steps

2 (lo g_{6} (x) - 2 lo g_{6} (z)) + 3 lo g_{6} (w)

Apply log rule:

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

2 lo g_{6} (z) = lo g_{6} (z^{2})

= 2 (lo g_{6} (x) - lo g_{6} (z^{2})) + 3 lo g_{6} (w)

Apply log rule:

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

lo g_{6} (x) - lo g_{6} (z^{2}) = lo g_{6} (\frac{x}{z ^{2}})

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

Apply log rule:

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

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

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

Apply log rule:

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

3 lo g_{6} (w) = lo g_{6} (w^{3})

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

Apply log rule:

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

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

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

Simplify

(\frac{x}{z ^{2}})^{2} w^{3} : \frac{x ^{2} w ^{3}}{z ^{4}}

= lo g_{6} (\frac{x ^{2} w ^{3}}{z ^{4}})

ln (\frac{x + 1}{9 - x ^{2}})

ln (\frac{3}{4}) - ln (2)

\frac{100}{2 \cdot π \cdot 200} \cdot ln (\frac{4 + 2 \cdot 20 0 ^{2}}{25})

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

ln (- 3 e^{2 x})

What is 2(log_{6}(x)-2log_{6}(z))+3log_{6}(w) ?
The solution to 2(log_{6}(x)-2log_{6}(z))+3log_{6}(w) is log_{6}((x^2w^3)/(z^4))