Start by applying the rule log_a(n) + log_a(m) = log_a(n xx m)loga(n)+loga(m)=loga(n×m).
ln(x(x - 2)) = 1ln(x(x−2))=1
ln(x^2 - 2x) = 1ln(x2−2x)=1
x^2 - 2x = e^1x2−2x=e1
x^2 - 2x = ex2−2x=e
x^2 - 2x - e = 0x2−2x−e=0
Solve by the quadratic formula.
x = (-(-2) +- sqrt(-2^2 - (4 xx 1 xx -e)))/(2 xx 1)x=−(−2)±√−22−(4×1×−e)2×1
x = (2 +- sqrt(4 + 4e))/2x=2±√4+4e2
x = (2 +- sqrt(4(1 + e)))/2x=2±√4(1+e)2
x = (2 +- 2sqrt(1 + e))/2x=2±2√1+e2
x = 1 +- sqrt(1 + e)x=1±√1+e
However, the -− solution is extraneous, since it renders the original equation undefined. So, the only actual solution is x = 1+sqrt(1 + e)x=1+√1+e
Hopefully this helps!
