Question

what is the true solution to $2 \ln 4x = 2 \ln 8$?
\\(\circ\\) $x = -4$
\\(\circ\\) $x = -2$
\\(\circ\\) $x = 2$
\\(\circ\\) $x = 4$

Explanation:

Step1: Divide both sides by 2

Given the equation \(2\ln(4x) = 2\ln(8)\), we first divide both sides by 2 to simplify. This gives us \(\ln(4x)=\ln(8)\).

Step2: Use the property of logarithms

Since the natural logarithm function \(y = \ln(x)\) is one - to - one (if \(\ln(a)=\ln(b)\), then \(a = b\) for \(a>0\) and \(b>0\)), we can set the arguments of the logarithms equal to each other. So we have the equation \(4x=8\).

Step3: Solve for x

To solve for \(x\), we divide both sides of the equation \(4x = 8\) by 4. \(\frac{4x}{4}=\frac{8}{4}\), which simplifies to \(x = 2\). We also need to check the domain of the original logarithmic equation. For \(\ln(4x)\) and \(\ln(8)\), the argument of the logarithm must be positive. When \(x = 2\), \(4x=8>0\), so \(x = 2\) is a valid solution.

Answer:

\(x = 2\) (corresponding to the option \(x = 2\))