Question

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

Answer 1

2

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:

`(1)/(2)\ln(64)=\ln(4x)`

Use power rule:

`\ln(64^(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.