# How do you use the properties of logarithms to expand ln((x^4sqrty)/z^5)?

Oct 22, 2017

$4 \ln x + \frac{1}{2} \ln y - 5 \ln z$

#### Explanation:

There are three properties of logarithms that will be useful here:

log_a (b/c) = log_a b - log_a c color(white)("aaaaaa")"Quotient Property"

log_a (b*c) = log_a b + log_a c color(white)("aaaaaa")"Product Property"

log_a (b^c) = c*log_a b color(white)("aaaaaaaaaaaaa")"Exponent Property"

Begin by using the Quotient Property to split apart the logarithm:

$\ln \left(\frac{{x}^{4} \sqrt{y}}{{z}^{5}}\right) = \ln \left({x}^{4} \sqrt{y}\right) - \ln \left({z}^{5}\right)$

Now, rewrite the square root as a fractional exponent:

$\ln \left({x}^{4} \sqrt{y}\right) - \ln \left({z}^{5}\right) = \ln \left({x}^{4} \cdot {y}^{\frac{1}{2}}\right) - \ln \left({z}^{5}\right)$

Now use the Product Property to split apart the first logarithm:

$\ln \left({x}^{4} \cdot {y}^{\frac{1}{2}}\right) - \ln \left({z}^{5}\right) = \ln {x}^{4} + \ln {y}^{\frac{1}{2}} - \ln {z}^{5}$

Lastly, use the Exponent Property to rewrite the logarithms:

$\ln {x}^{4} + \ln {y}^{\frac{1}{2}} - \ln {z}^{5} = 4 \ln x + \frac{1}{2} \ln y - 5 \ln z$