Question

How do you solve `log_10(x+4)-log_10x=log_10(x+2)`?

Answer 1

I got: `x=(-1+sqrt(17))/2`

Explanation:

We can use a property of logs to write a difference of logs into a log of a fraction:
`log_(10)((x+4)/(x))=log_(10)(x+2)`
If the logs are equal then also the arguments have to be. So:
`(x+4)/x=x+2`
Rearrange and solve for `x`:
`x+4=x^2+2x`
`x^2+x-4=0`
Use the Quadratic Formula:
`x_(1,2)=(-1+-sqrt(1+16))/2`
I can only accept the positive solution to avoid a negative argument in `log_(10)(x)`.