How do i know that an equation will have an extraneous solution?

Question

How do I know that an equation will have an extraneous solution?

For example in this question: 2log9(x) = log9(2) + log9(x + 24)

Answer 1

After you obtain the solution, substitute it in and check whether it is a valid solution.

For this particular question, a possible way to solve the problem is to bring the $2$ up and become the power and we might have included an extra solution.

$$x^2 = 2(x+24)$$

$$x^2-2x-48=0$$

$$(x-8)(x+6)=0$$

and we see that we need to exclude the negative solution.

Answer 2

First identify where each of the terms remains defined.

For real domain, we need $x,x+24>0\implies x>0$

$$\implies\log_9(x^2)=\log_92+\log_9(x+24)=\log_92(x+24)$$

$$\iff x^2=2(x+24)$$

Answer 3

If some step in your process of solving the equation was only a logical implication (rather than an equivalence), then the process may have introduced extraneous solutions, and you need to check all of them to be sure.

Typically the difference is that a step in the solution is not reversible. For example: if $a=b$ then we can conclude that $a^2=b^2$, but we cannot reliably conclude from $a^2=b^2$ that $a=b$ also holds. Similarly from $\log_ax=\log_a=y$ (with $a$ a positive constant $\neq1$) we can conclude that $x=y$, but from $x=y$ we cannot conclude that $\log_a x=\log_a y$ because both $x$ and $y$ could be negative, and outside the domain of definition.

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It is IMHO pointless to memorize rules, when you are supposed to think about the logic instead. Alas, teachers worldwide seem to prefer memorizing exceptions to thinking. Luckily my junior high teacher was of a different breed. That was before the people higher up started thinking that logic = elitism, and doing well in PISA tests is all students should aim at.