I use two rules which you should memorize.
First, if A and B are numbers greater than zero, then log A + log B = log AB.
This rule combines two logs into one.
Second, if log X = Y, then by definition, 10Y = X
This rule changes a log equation into an exponential equation.
So starting with log X + log (X + 15) = 2, we use the first rule to get
log [X(X + 15)] = 2
using the second rule, we get
102 = X(X + 15)
Now we can solve for X
X2 + 15X - 100 = 0
(X - 5) (X + 20) = 0
Therefore X = 5 or X = -20
The answer X = -20 is eliminated because if we use it in the original equation we get log (-20) which is undefined because the domain of the log function is X > 0.
So we have one answer X = 5