How do you use the properties of logarithms to expand #log_5 (x^2/(y^2z^3))#?

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Answer 1 · 2017-02-24T08:47:49

#2log_5(x) - 2*log_5(y) - 3*log_5(z)#

Property of log #1:

#log_x(a/b) = log_x(a)-log_y(b)#

So: #log_5(x^2/(y^2*z^3))=log_5(x^2)- log_5(y^2*z^3)#

Property of log #2:

#log_x(a*b) = log_x(a)+log_y(b)#

So: #log_5(y^2*z^3) = log_5(y^2) + log_5(z^3)#

Property of log #3:

#log_x (a^n) = n*log_x(a)#

So: #log_5(x^2) = 2log_5(x)#
#log_5(y^2)=2*log_5(y)#
#log_5(z^3) = 3*log_5(z)#

Final Answer:

#log_5(x^2/(y^2*z^3))= 2log_5(x) - [ 2*log_5(y) + 3*log_5(z)] = 2log_5(x) - 2*log_5(y) - 3*log_5(z)#

Hope this helps!