Question

Question #cae1a

Answer 1

`x=5`

Explanation:

Use the property `log(a)+log(b)=log(ab)` to simplify the equation to `log((x+1)*(x-1))=log(24)`.

This implies that `(x+1)*(x-1)=24`. Then, `x^2-1=24`. Add `1` to both sides: `x^2=25`. There are two solutions, `x=+-5`.

However, if we substitute `-5` back into the equation, we would be taking the logarithm of a negative number, which is not allowed. We can eliminate `-5` as an answer.

The only real solution is `5`.

Answer 2

`color(green)(x=5)`

Explanation:

Remember:
[1]`color(white)("XXX")log(a)+log(b)=log(a * b)`
[2]`color(white)("XXX")log(c)` is only defined for `c > 0`

`log(x+1)+log(x-1)`
`color(white)("XXX")=log(x^2-1) =log(24)`

`rarrx^2-1=24`

`rarr x^2=25`

`rarr x=+-5`

...but neither `log(x+1)` nor `log(x-1)` are defined if `x=-5`,
so this is an extraneous solution,

leaving only
`color(white)("XXX")x=5`