What is ln(a/b)-3ln(a^2)+ln(b^{-2}) ?

The solution to ln(a/b)-3ln(a^2)+ln(b^{-2}) is -ln(a^5b^3)

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simplify ln(a/b)-3ln(a^2)+ln(b^{-2})

simplify

ln (\frac{a}{b}) - 3 ln (a^{2}) + ln (b^{- 2})

- ln (a^{5} b^{3})

Solution steps

ln (\frac{a}{b}) - 3 ln (a^{2}) + ln (b^{- 2})

Apply log rule:

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

3 ln (a^{2}) = ln ((a^{2})^{3})

= ln (\frac{a}{b}) - ln ((a^{2})^{3}) + ln (b^{- 2})

Apply log rule:

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

ln (\frac{a}{b}) - ln ((a^{2})^{3}) = ln (\frac{\frac{a}{b}}{( a ^{2} ) ^{3}})

= ln (\frac{\frac{a}{b}}{( a ^{2} ) ^{3}}) + ln (b^{- 2})

Simplify

\frac{\frac{a}{b}}{( a ^{2} ) ^{3}} : \frac{1}{a ^{5} b}

= ln (\frac{1}{a ^{5} b}) + ln (b^{- 2})

Apply log rule:

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

ln (\frac{1}{a ^{5} b}) + ln (b^{- 2}) = ln (b^{- 2} \frac{1}{a ^{5} b})

= ln (b^{- 2} \frac{1}{a ^{5} b})

Simplify

\frac{1}{a ^{5} b} b^{- 2} : \frac{1}{a ^{5} b ^{3}}

= ln (\frac{1}{a ^{5} b ^{3}})

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (a^{5} b^{3})

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ln (\frac{a ^{2}}{b ^{4} c})

What is ln(a/b)-3ln(a^2)+ln(b^{-2}) ?
The solution to ln(a/b)-3ln(a^2)+ln(b^{-2}) is -ln(a^5b^3)