What is ln((2^{-3})/(3^25^{-2)}) ?

The solution to ln((2^{-3})/(3^25^{-2)}) is -3ln(2)-2ln(3)+2ln(5)

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simplify ln((2^{-3})/(3^25^{-2)})

simplify

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

- 3 ln (2) - 2 ln (3) + 2 ln (5)

Decimal Notation

- 1.05779 \dots

Solution steps

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

Apply log rule:

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

ln (\frac{2 ^{- 3}}{3 ^{2} \cdot 5 ^{- 2}}) = ln (2^{- 3}) - ln (3^{2} \cdot 5^{- 2})

= ln (2^{- 3}) - ln (5^{- 2} \cdot 3^{2})

Apply log rule:

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

ln (2^{- 3}) = - 3 ln (2)

= - 3 ln (2) - ln (3^{2} \cdot 5^{- 2})

Apply log rule:

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

ln (3^{2} \cdot 5^{- 2}) = ln (3^{2}) + ln (5^{- 2})

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

Apply log rule:

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

ln (3^{2}) = 2 ln (3)

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

Apply log rule:

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

ln (5^{- 2}) = - 2 ln (5)

= - 3 ln (2) - (2 ln (3) - 2 ln (5))

- (2 ln (3) - 2 ln (5)) : - 2 ln (3) + 2 ln (5)

= - 3 ln (2) - 2 ln (3) + 2 ln (5)

\frac{1}{3} lo g_{10} (27) - \frac{1}{2} lo g_{10} (25)

lo g_{4} (5 x^{3} y)

lo g_{2} (\frac{x ^{2}}{x + 1})

lo g_{10} (\frac{x}{7})

4 (- ln (e^{3} z^{\frac{5}{2}}) + ln (\frac{x ^{3}}{y ^{2}}))

What is ln((2^{-3})/(3^25^{-2)}) ?
The solution to ln((2^{-3})/(3^25^{-2)}) is -3ln(2)-2ln(3)+2ln(5)