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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (a^{1}) = 1 \cdot ln (a)

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

Apply log rule:

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

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

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

Apply log rule:

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

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

= 1 \cdot ln (a) - (- 3 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)

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

Multiply:

1 \cdot ln (a) = ln (a)

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

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

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

ln (3) + ln (x^{2})

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