Question

identify the solution set of (6 \ln e = e^{\ln 2x})

\\(\bigcirc\\) \\(\\{2\\}\\)

\\(\bigcirc\\) \\(\\{3\\}\\)

\\(\bigcirc\\) \\(\\{6\\}\\)

Explanation:

⚡ Using what you learned: Logarithms and Exponential Equations

Step 1: Simplify the left side

Using the property \( \ln(e) = 1 \):

$$ 6 \ln(e) = 6(1) = 6 $$

Step 2: Simplify the right side

Using the inverse property of logarithms and exponents, \( e^{\ln(u)} = u \):

$$ e^{\ln(2x)} = 2x $$

Step 3: Solve for x

Equate the simplified sides:

$$ 6 = 2x $$

$$ x = 3 $$

Answer:

{3}