A False Answer

Solve ln(x) + ln(x + 4) = ln(5). Join the left into ln(x(x + 4)), set the insides equal, and you reach x^2 + 4x - 5 = 0, which factors as (x + 5)(x - 1) = 0. The candidates are 1 and -5.

Try -5 in the joined equation and it seems fine: x(x + 4) is -5 times -1, which is 5. Try it in the original and it fails at once, since ln(-5) has no value. So -5 is a false answer, also called extraneous, and only x = 1 survives.

Where did it come from? The original needs each input positive: x and x + 4 both above 0. The joined log only needs their product positive, and two negatives make a positive too. Joining the logs loosened the rule, so the quadratic let in an answer the original never allowed. That is why the check always goes back to the original equation, never to a line you made.