What is expand ln(((e^5x^{-2})/(y^3))^6) ?

The solution to expand ln(((e^5x^{-2})/(y^3))^6) is 6(ln((e^5)/(x^2))-3ln(y))

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expand ln(((e^5x^{-2})/(y^3))^6)

expand

ln ((\frac{e ^{5} x ^{- 2}}{y ^{3}})^{6})

6 (ln (\frac{e ^{5}}{x ^{2}}) - 3 ln (y))

Solution steps

ln ((\frac{e ^{5} x ^{- 2}}{y ^{3}})^{6})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((\frac{e ^{5} x ^{- 2}}{y ^{3}})^{6}) = 6 ln (\frac{e ^{5} x ^{- 2}}{y ^{3}})

= 6 ln (\frac{e ^{5} x ^{- 2}}{y ^{3}})

Apply log rule:

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

ln (\frac{e ^{5} x ^{- 2}}{y ^{3}}) = ln (e^{5} x^{- 2}) - ln (y^{3})

= 6 (ln (e^{5} x^{- 2}) - ln (y^{3}))

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (y^{3}) = 3 ln (y)

= 6 (ln (e^{5} x^{- 2}) - 3 ln (y))

Multiply

e^{5} x^{- 2} : \frac{e ^{5}}{x ^{2}}

= 6 (ln (\frac{e ^{5}}{x ^{2}}) - 3 ln (y))

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What is expand ln(((e^5x^{-2})/(y^3))^6) ?
The solution to expand ln(((e^5x^{-2})/(y^3))^6) is 6(ln((e^5)/(x^2))-3ln(y))