Assume log base is 10.
log10(x(x-9))=1 is possibly another way of writing the equation.
x2-9x=101=10, x2-9x-10=0=(x-10)(x+1), so x=-1 or 10, but log(x) cannot be evaluated when x<0. So x=10 should be the solution.
log10(x)+log10(x-9)=1+log10(1)=1+0=1 checks out OK.