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

- 7.49554 \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 (3^{2} \cdot 5^{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)

ln (17) + ln (16)

\frac{6}{5} ln (1 - \frac{y}{x}) - \frac{1}{5} ln (\frac{y}{x} + 4)

- lo g_{10} (6.8 \cdot 1 0^{4})

- ln (x) + \frac{1}{2} ln (b) - ln (c)

ln (\frac{e ^{9}}{19})

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