Question

Solve: In(x-2)-In(x+2)=4: explain how you got to your answer.

Answer 1

No real solutions.

Explanation:

We want to solve the equation:

`\ln(x-2)+\ln (x+2)=4`

Recall that:

`\displaystyle \ln a-\ln b=\ln\left((a)/(b)\right)`

Therefore:

`\displaystyle \ln\left((x-2)/(x+2)\right)=4`

By Definition:

`\displaystyle e^4=(x-2)/(x+2)`

Cross-multiply:

`e^4(x+2)=x-2`

Distribute:

`xe^4+2e^4=x-2`

Isolate the x:

`xe^4-x=-2-2e^4`

Factor:

`\displaystyle x(e^4-1)=-2-2e^4`

Divide. Therefore:

`\displaystyle x=(-2-2e^4)/(e^4-1)=(2+2e^4)/(1-e^4)\approx-2.07`

However, note that logs cannot be negative and must be nonzero. According to the first logarithm, x > 2 and according to the second, x > -2. Since the answer is not greater than two, there are no real solutions.