Begin by moving both of the loglog terms to the left hand side.
log(x+8) - log(x-10) = 1log(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)log(mn)=log(m)−log(n)
Letting m=x+8m=x+8 and n=x-10n=x−10, we get;
log((x+8)/(x-10)) = 1log(x+8x−10)=1
Since we are working with a common loglog it is base ten. That means that the part inside of the parenthesis is equal to 1010 raised to the power of the right hand side, or;
10^1 = (x+8)/(x-10)101=x+8x−10
Now we just need to do some algebra to solve for xx. First, multiply both sides by (x-10)(x−10).
10(x-10) = x+810(x−10)=x+8
Now multiply the 1010 through the parenthesis.
10x - 100 = x + 810x−100=x+8
Subtract xx and add 100100 to both sides.
9x = 1089x=108
Finally, divide both sides by 99 to find xx.
x = 12x=12
