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

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Answer 1 · 2017-10-22T14:33:01

#4ln x + 1/2 ln y - 5 ln z#

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#