Put everything that's a log on the same side
log(x-15)+log(x) = 2log(x−15)+log(x)=2
Remember that log(m) + log(n) = log(mn)log(m)+log(n)=log(mn)
log(x(x-15)) = 2log(x(x−15))=2
If log_a(b) = cloga(b)=c, then b = a^cb=ac
x(x-15) = 10^(2)x(x−15)=102
Expand and solve the quadratic equation
x^2 - 15x = 100 rarr x^2 -15x -100 = 0x2−15x=100→x2−15x−100=0
x = (15 +-sqrt(225 - 4*1(-100)))/2 = (15 +-sqrt(225 +400))/2x=15±√225−4⋅1(−100)2=15±√225+4002
x = (15 +-sqrt(625))/2 = (15 +-25)/2x=15±√6252=15±252
x_1 = (15+25)/2 = 40/2 = 20x1=15+252=402=20
x_2 = (15-25)/2 = -10/2 = -5x2=15−252=−102=−5
Remember that since we were dealing with logarithms, we can't have null or negative arguments, so
x-15 > 0 rarr x > 15x−15>0→x>15
x > 0x>0
We conclude that any answers must follow x > 15x>15, which only of the two answers do, thus, the answer is x = 20x=20
