Solve \( -4 \ln(6x) = 2 \) Using Natural Logarithm Inverse Property - Math Solver AI

$Math Problem$

Mathematics High School

Question Content

Solve the equation \( -4 \ln(6x) = 2 \)

Correct Answer

A

Detailed Solution Steps

1
Isolate the logarithm: Divide both sides by -4: \( \ln(6x) = -\frac{1}{2} \)
2
Exponentiate with base \( e \): \( e^{\ln(6x)} = e^{-1/2} \)
3
Simplify: \( 6x = e^{-1/2} \)
4
Solve for \( x \): \( x = \frac{e^{-1/2}}{6} \approx \frac{0.6065}{6} \approx 0.101 \)

Knowledge Points Involved

1

Natural Logarithm (ln)

The logarithm with base \( e \) (≈2.71828), written as \( \ln(x) = \log_e(x) \).

2

Inverse Property of Logarithms

For \( a > 0 \), \( e^{\ln(a)} = a \), allowing elimination of natural logs by exponentiation.

3

Solving Linear Equations

Isolating the variable by performing inverse operations (e.g., division, exponentiation) to solve for \( x \).

Loading solution...