What is expand ln((a^{-2})/(b^{-4)c^5}) ?

The solution to expand ln((a^{-2})/(b^{-4)c^5}) is -2ln(a)+4ln(b)-5ln(c)

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expand ln((a^{-2})/(b^{-4)c^5})

expand

ln (\frac{a ^{- 2}}{b ^{- 4} c ^{5}})

- 2 ln (a) + 4 ln (b) - 5 ln (c)

Solution steps

ln (\frac{a ^{- 2}}{b ^{- 4} c ^{5}})

Apply log rule:

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

ln (\frac{a ^{- 2}}{b ^{- 4} c ^{5}}) = ln (a^{- 2}) - ln (b^{- 4} c^{5})

= ln (a^{- 2}) - ln (b^{- 4} c^{5})

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^{- 4} c^{5})

Apply log rule:

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

ln (b^{- 4} c^{5}) = ln (b^{- 4}) + ln (c^{5})

= - 2 ln (a) - (ln (c^{5}) + ln (b^{- 4}))

Apply log rule:

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

ln (b^{- 4}) = - 4 ln (b)

= - 2 ln (a) - (- 4 ln (b) + ln (c^{5}))

Apply log rule:

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

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

= - 2 ln (a) - (- 4 ln (b) + 5 ln (c))

- (- 4 ln (b) + 5 ln (c)) : 4 ln (b) - 5 ln (c)

= - 2 ln (a) + 4 ln (b) - 5 ln (c)

ln (12) - ln (2)

2 lo g_{3} (u) - lo g_{3} (v)

\frac{1}{4} lo g_{b} (x) + 2 lo g_{b} (y) - 4 lo g_{b} (x)

2 ln (x) - 5 ln (y)

lo g_{2} (\frac{x ^{5} 4 ^{x}}{2 ^{x^{4}}})

What is expand ln((a^{-2})/(b^{-4)c^5}) ?
The solution to expand ln((a^{-2})/(b^{-4)c^5}) is -2ln(a)+4ln(b)-5ln(c)