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We can start solving this question by bringing all terms to the same base. 10 in this case.
Let's use the identity log10(10) = 1
Substitute log10(10) instead of 1 on the left hand side,
log10(x+2) + log10x - log10(10) = log10(3/2)
now, combine the terms on the left hand side,
log10( (x+2)x/10 ) = log10(3/2)
(x + 2)x/10 = 3/2
multiply both sides by 10,
(x + 2)x = 15
x^2 + 2x - 15 = 0
(x + 5)(x - 3) = 0
x = 5
x = -5 and x = 3
Let me know if you have any questions.