Question 322465
{{{4ln(sqrt(x))-ln(xy^5)+3ln(y)}}} Start with the given expression.
 


{{{4ln(x^(1/2)^"")-ln(xy^5)+3ln(y)}}} Convert to exponential notation.



{{{ln(x^(4(1/2))^"")-ln(xy^5)+ln(y^3)}}} Use the identity  {{{y*ln(x)=ln(x^y)}}}



{{{ln(x^(4/2)^"")-ln(xy^5)+ln(y^3)}}} Multiply



{{{ln(x^2)-ln(xy^5)+ln(y^3)}}} Reduce.



{{{ln((x^2)/(xy^5))+ln(y^3)}}} Combine the first two logs using the identity {{{ln(A)-ln(B)=ln(A/B)}}}



{{{ln(((x^2)/(xy^5))(y^3))}}} Combine the logs using the identity {{{ln(A)+ln(B)=ln(A*B)}}}



{{{ln((x^2y^3)/(xy^5))}}} Combine the fractions.



{{{ln(x/(y^2))}}} Reduce.



So {{{4ln(sqrt(x))-ln(xy^5)+3ln(y)=ln(x/(y^2))}}}