log10(x+2) + log10(x) - 1 = log10(3/2)

Question

How do you solve this equation for x?

I tried the following

log10(x+2) + log10(x) - 1 = log10(3/2)

log10(x+2) + log10(x) - log10(10) = log10(3/2)

log10((x+2)*x / 10) = log10(3/2)

(x+2)*x / 10 = 1.5

(x+2)*x = 15

x^2 + 2x - 15 = 0

(x+5)*(x-3) = 0

x1 = 3 and x2 = -5

But my math book gives only the answer x=3

There are a few equation solver (one is WolframAlpha) which also give the answer x=3

There are a few other equation solver with x1=3 and x2 = -5

(And there are also equation solver with x = 0.25)

Now my question: Is x2 = -5 a forbidden solution (I can't see why since I don't divide by -5) or are there any mistakes in my calculations?

Answer 1

If this is the problem........log₁₀(x+2) + log₁₀(x) - 1 = log₁₀(1.5)

Then log₁₀{(x+2)(x)} - 1 = log₁₀(1.5)

Then log₁₀{(x+2)(x)} - log₁₀(1.5) = 1 or log₁₀{(x+2)(x)/(1.5)} = 1

Take each side to the 10th power to get

(x+2)(x)/(1.5) =10¹ then (x+2)(x) =15 or x² + 2x = 15 or x²+2x -15 = 0

This factors to (x+5)(x-3) = 0. So x = -5 & 3

The problems is that you cannot take the log of a negative number so 3 is the only realistic solution!