What is 4(log_{5}(y)-2log_{5}(z))+3log_{5}(w) ?

The solution to 4(log_{5}(y)-2log_{5}(z))+3log_{5}(w) is log_{5}((y^4w^3)/(z^8))

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simplify 4(log_{5}(y)-2log_{5}(z))+3log_{5}(w)

simplify

4 (lo g_{5} (y) - 2 lo g_{5} (z)) + 3 lo g_{5} (w)

lo g_{5} (\frac{y ^{4} w ^{3}}{z ^{8}})

Solution steps

4 (lo g_{5} (y) - 2 lo g_{5} (z)) + 3 lo g_{5} (w)

Apply log rule:

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

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

= 4 (lo g_{5} (y) - lo g_{5} (z^{2})) + 3 lo g_{5} (w)

Apply log rule:

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

lo g_{5} (y) - lo g_{5} (z^{2}) = lo g_{5} (\frac{y}{z ^{2}})

= 4 lo g_{5} (\frac{y}{z ^{2}}) + 3 lo g_{5} (w)

Apply log rule:

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

4 lo g_{5} (\frac{y}{z ^{2}}) = lo g_{5} ((\frac{y}{z ^{2}})^{4})

= lo g_{5} ((\frac{y}{z ^{2}})^{4}) + 3 lo g_{5} (w)

Apply log rule:

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

3 lo g_{5} (w) = lo g_{5} (w^{3})

= lo g_{5} ((\frac{y}{z ^{2}})^{4}) + lo g_{5} (w^{3})

Apply log rule:

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

lo g_{5} ((\frac{y}{z ^{2}})^{4}) + lo g_{5} (w^{3}) = lo g_{5} ((\frac{y}{z ^{2}})^{4} w^{3})

= lo g_{5} ((\frac{y}{z ^{2}})^{4} w^{3})

Simplify

(\frac{y}{z ^{2}})^{4} w^{3} : \frac{y ^{4} w ^{3}}{z ^{8}}

= lo g_{5} (\frac{y ^{4} w ^{3}}{z ^{8}})

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\frac{1}{2} ln (5 + 2) - \frac{1}{2} ln (5 + 0)

ln (\frac{1}{( 3.5 e ^{(- 4 T)} - 5 e ^{(- T)} + 2.5 )})

lo g_{b} (\frac{a ^{2} 2}{e ^{3}})

ln (v^{- \frac{1}{3}}) + ln ((v - 3)^{\frac{1}{3}}) + c

What is 4(log_{5}(y)-2log_{5}(z))+3log_{5}(w) ?
The solution to 4(log_{5}(y)-2log_{5}(z))+3log_{5}(w) is log_{5}((y^4w^3)/(z^8))