How do you use the properties of logarithms to expand #lnroot4(x^3(x^2+3))#?

1 Answer
Douglas K.
Aug 28, 2017

Given: #ln(root4(x^3(x^2+3)))#

The root 4 can be written as the #1/4# power:
#ln((x^3(x^2+3))^(1/4))#

The property #ln(a^c) = cln(a)# tells us that the #1/4# comes outside as multiplication:

#1/4ln(x^3(x^2+3))#

Use the property #ln(uv) = ln(u) + ln(v)# to make the two factors within the argument become the sum of two logarithms but both are still multiplied by #1/4#:

#1/4ln(x^3)+1/4ln(x^2+3)#

Use the property #ln(a^c) = cln(a)# to bring the cube outside as multiplication by 3:

#3/4ln(x)+1/4ln(x^2+3)#

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