Expand this logarithmic equation (using properties) : #ln(sqrt(x-5)/y^2)# how do i do this?

Question

this is what my teacher did but i think she did it wrong and i feel like the #1/5# is supposed to be #1/2#. someone pls help and tell me if im wrong or right!

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Answer 1 · 2017-12-21T01:06:24

You are correct that it should be #1/2#:

#ln(sqrt(x-5)/y^2) = (1/2)ln(x-5)-(2)ln(y)#

Given: #ln(sqrt(x-5)/y^2)#

Use the property #ln(a/b) = ln(a) - ln(b)# where #a = sqrt(x-5) and #b = y^2#:

#ln(sqrt(x-5)/y^2) = ln(sqrt(x-5))-ln(y^2)#

Write #sqrt(x-5)# as #(x-5)^(1/2)#:

#ln(sqrt(x-5)/y^2) = ln((x-5)^(1/2))-ln(y^2)#

Use the property #ln(a^c) = (c)ln(a)# on the first term:

#ln(sqrt(x-5)/y^2) = (1/2)ln(x-5)-ln(y^2)#

Do the same on the second term:

#ln(sqrt(x-5)/y^2) = (1/2)ln(x-5)-(2)ln(y)#