Question

what is the true solution to $2 \ln e^{\ln 5x} = 2 \ln 15$?
$\bigcirc$ $x = 0$
$\bigcirc$ $x = 3$
$\bigcirc$ $x = 9$
$\bigcirc$ $x = 15$

Explanation:

Step1: Simplify left side using log property

Recall that $\ln e^a = a$. So for $2\ln e^{\ln 5x}$, first simplify $\ln e^{\ln 5x}=\ln 5x$. Then the left side becomes $2\ln 5x$. The equation is now $2\ln 5x = 2\ln 15$.

Step2: Divide both sides by 2

Divide each side of the equation by 2: $\frac{2\ln 5x}{2}=\frac{2\ln 15}{2}$, which simplifies to $\ln 5x=\ln 15$.

Step3: Use one - to - one property of logs

If $\ln a=\ln b$, then $a = b$ (for $a>0,b>0$). So from $\ln 5x=\ln 15$, we get $5x = 15$.

Step4: Solve for x

Divide both sides of $5x = 15$ by 5: $x=\frac{15}{5}=3$.

Answer:

x = 3 (the option corresponding to x = 3, which is the second option: x = 3)