How should I be avoiding this mistake? (To avoid missing solutions)

Question

First of all, I am sorry if this is a question too simple or stupid.

Consider the equation:

$$ \log((x+2)^2) = 2 \log(5) $$

If I apply the logarithm law $ \log_a(b^c) = c \log_a(b) $

$$ \begin{align} 2 \log(x+2) & = 2 \log(5) \\ \log(x+2) &= \log(5) \\ x+2 &= 5 \\ x &= 3 \end{align} $$

But I can see that I am missing a solution, $x = -7$. I noticed that

$$ \begin{align} \log((x+2)^2) &= 2 \log(5) \\ \Updownarrow \\ 2 \log(x+2) &= 2 \log(5) \end{align} $$

Is NOT true. The domain of the first equation is $x \in \mathbb{R}$ but the second equation's is $x \geq -2$.

I know the correct solution.

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So I understand that this is not an equivalent transformation of the equation. What I don't know is how I should avoid this. Is there something to keep in mind that would help me evade this mistake? Naturally, I wouldn't have noticed the missing solution, unless I checked the domain of the second equation, which I wouldn't really have had a reason for...

Answer 1

The formula $$\log_a b^c = c \log_a b$$ is true only if $b > 0$ (if we assume that $\log_a$ is a real-valued function). Therefore, an alternative method of solution can proceed as follows: $$\log (x+2)^2 = 2 \log 5 = \log 5^2 = \log 25,$$ and because now all the arguments to $\log$ on both sides must be positive, we have $$(x+2)^2 = 25$$ or $$(x+2-5)(x+2+5) = (x-3)(x+7) = 0,$$ and both solutions are found.

Answer 2

Rule of thumb when dealing with squares (even powers) if you don't have an absolute value you're most likely missing it.

Here as written in comments $\ln(x^2)=2\ln(|x|)$ because as you said their domains must be the same. This is because $\ln(ab)=\ln a+\ln b$ only if $a,b>0$.

Another example is that $\sqrt{x^2}=|x|$.

Answer 3

Congratulations on understanding why the other solution exists and why you missed it.

How to avoid this:

Whenever you see a real number squared, or raised to any even power, (make sure it's not a complex number that you're dealing with!), perform the following substitution:

$$x^2 \iff |x|^2$$

Note that both expressions are always equal (in real numbers), so it is 100% correct to make this substitution at any time.

Only then, perform your rules.

In other words, never "take the square out" unless it's surrounded by the absolute-value function.

In your example, $(x+2)^2$ would first become $|x+2|^2$, and only then you should proceed as you did, and this way you wouldn't miss any solutions.