Question

What is the solution to 3 In 2 +In 8=2 In(4x)

Answer 1

Answer:

2

Step-by-step explanation:

We want to condense left hand side into one logarithm

Use power rule:

$\ln(2^3)+\ln(8)=2 \ln(4x)$

Use product rule:

$\ln(2^3 \cdot 8)=2\ln(4x)$

Simplify:

$\ln(64)=2\ln(4x)$

Divide both sides by 2:

$\frac{1}{2}\ln(64)=\ln(4x)$

Use power rule:

$\ln(64^\frac{1}{2})=\ln(4x)$

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

$8=4x$

Divide both sides by 4:

$2=x$

$x=2$

Check:

$\ln(2^3)+\ln(8)=2 \ln(4x)$

Replace $x$ with 2:

$\ln(2^3)+\ln(8)=2 \ln(4(2))$

$\ln(64)=2 \ln(8)$

$\ln(64)=\ln(8^2)$

$\ln(64)=\ln(64)$ is true so $x=2$ is a solution.