Question

How do you solve `log (x + 8) = 1 + log (x - 10)`?

Answer 1

`x = 12`

Explanation:

Begin by moving both of the `log` terms to the left hand side.

`log(x+8) - log(x-10) = 1`

Now we can use the division rule for logarithms to combine both terms into one. The division rule states that;

`log(m/n) = log(m) - log(n)`

Letting `m=x+8` and `n=x-10`, we get;

`log((x+8)/(x-10)) = 1`

Since we are working with a common `log` it is base ten. That means that the part inside of the parenthesis is equal to `10` raised to the power of the right hand side, or;

`10^1 = (x+8)/(x-10)`

Now we just need to do some algebra to solve for `x`. First, multiply both sides by `(x-10)`.

`10(x-10) = x+8`

Now multiply the `10` through the parenthesis.

`10x - 100 = x + 8`

Subtract `x` and add `100` to both sides.

`9x = 108`

Finally, divide both sides by `9` to find `x`.

`x = 12`