Question

solve \\(ln 5 + ln 2x = 3\\).
(1 point)
\\(\bigcirc\\) \\(x = \dfrac{e^3 - 5}{2}\\)
\\(\bigodot\\) \\(x = \dfrac{e^3}{10}\\)
\\(\bigcirc\\) \\(x = \dfrac{3}{ln 10}\\)
\\(\bigcirc\\) \\(x = \dfrac{3 - ln 5}{2}\\)

Explanation:

Step1: Use logarithm addition rule

The sum of logarithms \(\ln a+\ln b = \ln(ab)\), so \(\ln 5+\ln 2x=\ln(5\times2x)=\ln(10x)\).
The equation becomes \(\ln(10x) = 3\).

Step2: Convert logarithmic to exponential form

Recall that if \(\ln y=k\), then \(y = e^{k}\). So for \(\ln(10x)=3\), we have \(10x=e^{3}\).

Step3: Solve for \(x\)

Divide both sides of the equation \(10x = e^{3}\) by 10: \(x=\frac{e^{3}}{10}\).

Answer:

\(x=\frac{e^{3}}{10}\) (corresponding to the option with \(x = \frac{e^{3}}{10}\))