How can $x=1$ be a solution to this equation?

Question

This is the equation I am supposed to solve: $$\log_{3}(2x^{2} - x-1)\ -\ \log_{3}(\ x-1)\ = 2$$ The textbook gives the solutions, $x=1, x=4$, with the working out as shown below:

While I understand what they are doing, I don't completely understand why they work it out as they do. Specifically, why isn't the term on the left side in the second row simplified? In my working out, $$\log_{3}\left(\frac{2x^{2} - x-1}{x-1}\right) \implies \log_{3}\left(\frac{(2x+1)(x-1)}{x-1}\right) \implies \log_{3}(2x+1)$$ This equals $2$, allowing it to be simplified further, \begin{align*} \: & \log_{3}(2x+1) = 2 \\ \implies \: & 2x+1 = 3^2 = 9 \\ \implies \: & 2x = 8 \\ \implies \: & x = 4 \\ \end{align*}

What's more, $x$ can't even equal $1$, because if it does, both logs simplify to $log(0)$, which is undefined. I am wondering if there is some other concept I am not aware of, or is the book making a mistake?

Answer 1

$x=1$ cannot be a solution because the equation is undefined for $x=1$. The equation in reals is only meaningful if $x-1 > 0$ and $2x^2-x-1>0$, which means $x>1$. In that case one may rewrite the equation as follows:

$$\begin{align} \log_3(2x^2-x-1) - \log_3(x-1) &= \log_3\big((2x+1)(x-1)\big) - \log_3(x-1) \\ &= \log_3(2x+1) + \log_3(x-1) - \log_3(x-1) \\ &= \log_3(2x+1) \\ &\stackrel!= 2 \end{align}$$ and thus $2x+1=9$ which has only one solution $x=4$.

As you see, the culprit is not a division by $x-1$ as clained in the comments, because it's still invalid if no division is present like above.

Answer 2

You are right, $x=1$ is not a solution. Note that:

$$\log_ca-\log_cb=\log_c\left(\frac ab\right)$$

Holds iff. $0<c\neq 1$ and $a,b>0$.

This phenomenon is called the Extraneous root in mathematics.

Answer 3

The issue of whether the book is technically correct is (unfortunately) a subjective communication issue.

Technically, the statement $x=1$ or $x=4$ can be interpreted as implying that any value of $x$ that satisfies the equation must be an element of $\{1,4\}$.

Therefore, the set $\{1,4\}$ represents the two candidate values that must each be individually explored to determine which of them (if any) solve the problem.

So, under this interpretation, the book is merely saying that it is impossible for any value of $x$ to solve the problem if that value of $x$ is not an element in $\{1,4\}.$

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Unfortunately, it is generally commonly understood that the naked statement $x=1$ or $x=4$ usually connotes that these are the answers, rather than that these are the candidate values.

So, the author was lazy, and just about any Math student (including me) would have looked at that and said : "on what planet"?

Answer 4

Technically, it is correct to write that \begin{align}&\log_3 (2x^2 - x-1)- \log_3(x-1) = 2\\\implies{}& x=1 \quad\text{OR}\quad x=4.\end{align} On the other hand, it would be incorrect to claim that the equation $$\log_3 (2x^2 - x-1)- \log_3(x-1) = 2$$ has solutions $1$ and $4,$ that is, that \begin{align}&\log_3 (2x^2 - x-1)- \log_3(x-1)\ = 2\\\iff{}& x=1 \quad\text{OR}\quad x=4.\end{align}

So, are you saying the book is technically correct? Maybe you could explain more in an answer?

1. Yes, albeit incomplete. If the exercise is to solve that given equation and you've completely reproduced your book's answer, then it can be finished by filling in the missing final step: pruning out any extraneous solution, in this case $x=1$ as you've pointed out.

   Noting the attendant implicit conditions $$a\in(0,1)\cup(1,\infty)\quad\text{and}\quad x>0$$ of $\log_ax$ can help eliminate extraneous solutions.

2. Alternatively (as you can see, the preceding pruning method is easier), its solution can be modified by carefully adapting the ⟹ to ⟺; here's a very detailed example: \begin{align}&\log_3 (2x^2 - x-1)- \log_3(x-1) = 2\\\iff{}&\log_3 \frac{2x^2 - x-1}{x-1} = 2 \quad\text{AND}\quad2x^2 - x-1>0\quad\text{AND}\quad x-1>0\\\iff{}&(\log_3(2x+1) = 2 \quad\text{AND}\quad x\ne1)\quad\text{AND}\quad2x^2 - x-1>0\quad\text{AND}\quad x-1>0\\ \iff{}& 2x+1= 9\quad\text{AND}\quad2x^2 - x-1>0\quad\text{AND}\quad x-1>0\\\iff{}& x=4.\end{align}