logx-log(x−10)=1
Recall that loga-logb=log(a/b), so
logx-log(x−10)=log(x/(x-10))
log(x/(x-10))=1
Convert the logarithmic equation to an exponential equation.
10^(log(x/(x-10))) = 10^1
Remember that 10^logx =x, so
x/(x-10)=10
x=10(x-10)
x=10x-100
9x=100
x=100/9
Check:
logx-log(x−10)=1
If x=100/9,
log(100/9)-log(100/9−10)=1
log(100/9)-log(100/9-90/9)=1
log(100/9)-log((100-90)/9)=1
log(100/9)-log(10/9)=1
log((100/color(red)(cancel(color(black)(9))))/(10/color(red)(cancel(color(black)(9)))))=1
log(100/10)=1
log10 = 1
1=1
x=100/9 is a solution.
