you want to use the product rule for logs on the left-hand side:
That would give you: log(x*(x+5)) = log 6
you can now "cancel out the logs" and you have x(x+5) = 6 ---> x^2 + 5x =6
Subtract 6 on both sides --> x^2 + 5x - 6 = 0
You're looking for 2 integers that multiply to give -6 while adding up to 5. That's 6 and -1, so you can factor this quadratic equation as follows:
(x+6)* (x-1) = 0 ---> x = -6 or x = 1
Those are solutions of the quadratic equation but you still have to check them in the original equation:
- starting with -6: log(-6) + log(-6+5) = log6 ? DOES NOT WORK because you cannot take the log of a negative number.
- checking x=1: log(1) + log(1+5) = log6 ? this one works!
So the only solution is x = 1