Question

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

Explanation:

Step1: Simplify left - hand side using logarithm property

Recall the property of logarithms: $\ln e^{a}=a$. For the left - hand side of the equation $2\ln e^{\ln 5x}$, first, simplify the inner part $\ln e^{\ln 5x}$. By the property $\ln e^{a}=a$, we have $\ln e^{\ln 5x}=\ln 5x$. So the left - hand side becomes $2\ln 5x$. The original equation $2\ln e^{\ln 5x}=2\ln 15$ is now $2\ln 5x = 2\ln 15$.

Step2: Divide both sides by 2

Divide both sides of the equation $2\ln 5x = 2\ln 15$ by 2. We get $\ln 5x=\ln 15$.

Step3: Use the one - to - one property of logarithms

If $\ln a=\ln b$, then $a = b$ (since the natural logarithm function $y = \ln x$ is one - to - one). So from $\ln 5x=\ln 15$, we can conclude that $5x = 15$.

Step4: Solve for x

Divide both sides of the equation $5x = 15$ by 5. We have $x=\frac{15}{5}=3$.

Answer:

B. $x = 3$