Question

what is the true solution to (2 ln 4x = 2 ln 8)?

• (x = -4)
• (x = -2)
• (x = 2)
• (x = 4)

Explanation:

⚡ Using what you learned: Logarithms and Exponential Equations

Step 1: Simplify the equation

Divide both sides of the equation by \( 2 \):

$$ \ln(4x) = \ln(8) $$

Step 2: Equate the arguments

Since the natural logarithm function is one-to-one, we can set the arguments equal to each other:

$$ 4x = 8 $$

Step 3: Solve for x

Divide both sides by \( 4 \):

$$ x = 2 $$

Step 4: Verify the solution

Check if \( x = 2 \) is in the domain of the original logarithmic expression (the argument must be positive):

$$ 4x = 4(2) = 8 > 0 $$

Since the argument is positive, \( x = 2 \) is a true solution.

Answer:

\( x = 2 \)