What is ln((a^{-2}b^3)/(c^{-5)}) ?

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

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simplify ln((a^{-2}b^3)/(c^{-5)})

simplify

ln (\frac{a ^{- 2} b ^{3}}{c ^{- 5}})

- 2 ln (a) + 3 ln (b) - - 5 ln (c)

Solution steps

ln (\frac{a ^{- 2} b ^{3}}{c ^{- 5}})

Apply log rule:

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

ln (\frac{a ^{- 2} b ^{3}}{c ^{- 5}}) = ln (a^{- 2} b^{3}) - ln (c^{- 5})

= ln (a^{- 2} b^{3}) - ln (c^{- 5})

Apply log rule:

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

ln (c^{- 5}) = - 5 ln (c)

= ln (a^{- 2} b^{3}) - - 5 ln (c)

Apply log rule:

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

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

= ln (a^{- 2}) + ln (b^{3}) - - 5 ln (c)

Apply log rule:

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

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

= - 2 ln (a) + ln (b^{3}) - - 5 ln (c)

Apply log rule:

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

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

= - 2 ln (a) + 3 ln (b) - - 5 ln (c)

\frac{1}{2} lo g_{2} (13)

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lo g_{12} (+ \frac{1}{2} lo g_{7} (- lo g_{2} (x)))

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