which of the following logarithmic expressions are equivalent to ln sqrt xy +ln (x/e)? select all that applya. ln (sqrt x^3 y)/e) b. ln (x^2y /2e)c. 1/2 ln x^2y-ed. ln(x sqrt xy)-1e. 3/2ln x+1/2ln y-1

Answer: Option d.Step-by-step explanation: It is important to remember the following proporties in order to solve the exercise: [tex]ln(a)+ln(b)=ln(ab)\\\\ln(a)-ln(b)=ln(\frac{a}{b})\\\\ln(a)^m=mln(a)\\\\ln(e)=1[/tex] In this case you have this expression given in the exercise: [tex]ln(\sqrt{xy})+ln(\frac{x}{e})[/tex] Since equivalent expression have always the same value, you can simplify the expression given in order to find an equivalent expression. Therefore, applying the corresponding properties, you get the following equivalent expression (Remember that [tex]\sqrt[n]{a^n}=a[/tex]): [tex]ln(\sqrt{xy})+ln(\frac{x}{e})=ln(\sqrt{xy})+ln(x)-ln(e)=ln(x\sqrt{xy})-1[/tex]