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

1 Answer
Oct 22, 2017

4ln x + 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 ((x^4 sqrt(y))/(z^5)) = ln (x^4 sqrt(y)) - ln (z^5)

Now, rewrite the square root as a fractional exponent:

ln (x^4 sqrt(y)) - ln (z^5) = ln (x^4 * y^(1/2)) - ln (z^5)

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

ln (x^4 * y^(1/2)) - ln (z^5) = ln x^4 + ln y^(1/2) - ln z^5

Lastly, use the Exponent Property to rewrite the logarithms:

ln x^4 + ln y^(1/2) - ln z^5 = 4ln x + 1/2 ln y - 5 ln z