Question

identify the solution set of (3 ln 4 = 2 ln x).

\\(\bigcirc\\) \\(\\{6\\}\\)
\\(\bigcirc\\) \\(\\{-8, 8\\}\\)
\\(\bigcirc\\) \\(\\{8\\}\\)

Explanation:

⚡ Using what you learned: Logarithms and Exponential Equations

Step 1: Apply the power property of logarithms

$$ \ln(4^3) = \ln(x^2) $$

$$ \ln(64) = \ln(x^2) $$

Step 2: Equate the arguments

$$ 64 = x^2 $$

Step 3: Solve for x and check restrictions

$$ x = \pm 8 $$

Since the original equation contains \(\ln x\), the domain requires \(x > 0\). Therefore, \(x = -8\) is extraneous.

$$ x = 8 $$

Answer:

{8}