# How do you expand (ln(5^1))(ln(2/(3^-2)^-2)?

Dec 15, 2015

$\ln \left(5\right) \left(\ln \left(2\right) - 4 \ln \left(3\right)\right)$

#### Explanation:

Begin by simplifying the second factor. We can distribute the negative exponent in the denominator as such:

$\ln \left({5}^{1}\right) \ln \left(\frac{2}{{3}^{-} 2} ^ - 2\right) = \ln \left(5\right) \ln \left(\frac{2}{3} ^ 4\right)$

Next we can use the quotient property of logarithms to expand the second term:

$\ln \left(5\right) \ln \left(\frac{2}{3} ^ 4\right) = \ln \left(5\right) \left(\ln \left(2\right) - \ln \left({3}^{4}\right)\right)$

Finally, we can use the power property logarithms to move the negative exponent outside the logarithm:

$\ln \left(5\right) \left(\ln \left(2\right) - \ln \left({3}^{4}\right)\right) = \ln \left(5\right) \left(\ln \left(2\right) - 4 \ln \left(3\right)\right)$