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

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

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simplify ln((a^{-4})/(b^{-4)c^4})

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (c^{4}) = 4 ln (c)

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

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

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

lo g_{2} (7 x)

ln^{2} (t) + 2 ln (t)

ln (x^{7} \cdot \frac{( x - 2 )}{2 x + 3})

\frac{1}{2} (ln (3^{2}) - ln (5^{4}) - ln (2^{2}))

lo g_{10} (z^{45})

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