Question

Lnx+Ln(2x)=1/e
Can someone help me solve this problem?

Answer 1

`D:x > 0\\\\ln(x)+ln(2x)=(1)/(e)\ \ \ \ |use\ ln(a)+ln(b)=ln(ab)\\\\ln(x\cdot2x)=(1)/(e)\\\\ln(2x^2)=(1)/(e)\ \ \ \ |use\ b=ln(e^b)\\\\ln(2x^2)=ln\left(e^(1)/(e)\right)\iff2x^2=e^(1)/(e)\ \ \ \ |divide\ both\ sides\ by\ 2\\\\x^2=(e^(1)/(e))/(2)\iff x=\sqrt(e^(1)/(e))/(2)\\\\x=\frac{\sqrt{e^(1)/(e)}}{\sqrt2}\cdot(\sqrt2)/(\sqrt2)\\\\\boxed{x=\frac{\sqrt{2e^(1)/(2)}}{2}}`