What is ln((a^{-3})/(b^{-1)c^3}) ?

The solution to ln((a^{-3})/(b^{-1)c^3}) is -3ln(a)+ln(b)-3ln(c)

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simplify ln((a^{-3})/(b^{-1)c^3})

simplify

ln (\frac{a ^{- 3}}{b ^{- 1} c ^{3}})

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

Solution steps

ln (\frac{a ^{- 3}}{b ^{- 1} c ^{3}})

Apply log rule:

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

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

= ln (a^{- 3}) - ln (b^{- 1} c^{3})

Apply log rule:

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

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

= - 3 ln (a) - ln (b^{- 1} c^{3})

Apply log rule:

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

ln (b^{- 1} c^{3}) = ln (b^{- 1}) + ln (c^{3})

= - 3 ln (a) - (ln (c^{3}) + ln (b^{- 1}))

Apply log rule:

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

ln (b^{- 1}) = - 1 \cdot ln (b)

= - 3 ln (a) - (- 1 \cdot ln (b) + ln (c^{3}))

Apply log rule:

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

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

= - 3 ln (a) - (- 1 \cdot ln (b) + 3 ln (c))

Multiply:

1 \cdot ln (b) = ln (b)

= - 3 ln (a) - (3 ln (c) - ln (b))

- (- ln (b) + 3 ln (c)) : ln (b) - 3 ln (c)

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

2 ln (3 + 2)

- lo g_{10} (2.5 \cdot 1 0^{- 9})

- lo g_{10} (2 \cdot 1 0^{- 7})

3 ln (x) + 2 ln (y) - \frac{1}{2} ln (x + y)

ln (x) - ln (\frac{y}{x})

What is ln((a^{-3})/(b^{-1)c^3}) ?
The solution to ln((a^{-3})/(b^{-1)c^3}) is -3ln(a)+ln(b)-3ln(c)