Question

which of the following correctly solves the equation $ln(3) = ln(12 - 12x)$? (1 point)
$x = \frac{4}{3}$
$x = \frac{5}{4}$
$x = -\frac{3}{4}$
$x = \frac{3}{4}$

Explanation:

Step1: Use the property of logarithms

If \(\ln(a)=\ln(b)\), then \(a = b\) (for \(a>0\) and \(b>0\)). So we set \(3=12 - 12x\).

Step2: Solve for \(x\)

Subtract 12 from both sides: \(3-12=12-12x - 12\), which simplifies to \(- 9=-12x\).
Divide both sides by \(- 12\): \(x=\frac{-9}{-12}=\frac{3}{4}\). We also need to check the domain: for \(\ln(3)\), the argument is 3>0; for \(\ln(12 - 12x)\), when \(x = \frac{3}{4}\), \(12-12\times\frac{3}{4}=12 - 9 = 3>0\), so it is valid.

Answer:

\(x=\frac{3}{4}\) (corresponding to the option \(x=\frac{3}{4}\))