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

1 Answer
Jul 2, 2018

Answer:

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

Explanation:

#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#