How do you use the properties of logarithms to expand #lnroot3(x^2/y^3)#?

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Answer 1 · 2018-07-02T19:56:20

#color(blue)(2/3lnx-lny)#

#root(3)(x^2/y^3)=(x^2/y^3)^(1/3)#

By the laws of logarithms:

#lna^b=blna#

#ln(a/b)=lna-lnb#

Hence:

#ln(x^2/y^3)^(1/3)=1/3ln(x^2/y^3)=1/3ln(x^2)-1/3ln(y^3)#

# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \=2/3lnx-3/3lny#

# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \=2/3lnx-lny#