What is ln((2^{-2})/(3^15^4)) ?

The solution to ln((2^{-2})/(3^15^4)) is -2ln(2)-ln(3)-4ln(5)

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simplify ln((2^{-2})/(3^15^4))

simplify

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

- 2 ln (2) - ln (3) - 4 ln (5)

Decimal Notation

- 8.92265 \dots

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (3^{1} \cdot 5^{4}) = ln (3^{1}) + ln (5^{4})

= - 2 ln (2) - (ln (3^{1}) + ln (5^{4}))

Apply log rule:

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

ln (3^{1}) = 1 \cdot ln (3)

= - 2 ln (2) - (1 \cdot ln (3) + ln (5^{4}))

Apply log rule:

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

ln (5^{4}) = 4 ln (5)

= - 2 ln (2) - (1 \cdot ln (3) + 4 ln (5))

Multiply:

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

= - 2 ln (2) - (4 ln (5) + ln (3))

- (ln (3) + 4 ln (5)) : - ln (3) - 4 ln (5)

= - 2 ln (2) - ln (3) - 4 ln (5)

\frac{3 ln ( \frac{2}{9} )}{4}

ln (\frac{67}{6})

ln (3 x^{3})

ln (\frac{2}{e}) + 1

ln (162) + ln (3) - 4 ln (3)

What is ln((2^{-2})/(3^15^4)) ?
The solution to ln((2^{-2})/(3^15^4)) is -2ln(2)-ln(3)-4ln(5)