Question

How do you solve `log _x 8 = -3`?

Answer 1

`x=1/2`

Explanation:

`log_x 8=-3`
`=>log_x 2^3=-3`
`3log_x 2=-3` (The logarithm of the `3^(rd)` power of a number is `3` times the logarithm of the number itself)
Divide both side into `1/3`, in order to simplify left side:
`1/3*3log_x 2=1/3*(-3)`
`=>log_x 2=-1`
`=>x^-1=2` (definition of logarithm)
Multiply both side by `x`:
`x*x^-1=x*2`
`=>2x=x^0` (If we take the product of two exponentials with the same nonzero base, we simply add the exponents)
Multiply both side by `1/2`, in order to simplify left side:
`1/2*2x=1/2*1`
`x=1/2`