solve $(3x-1)^{x-1}=1$

Question

Q:How do I solve for $x$? $$(3x-1)^{x-1}=1$$

When I first saw this question I thought it was quadratic equation, $(3x-1)(x-1)=1$. But friend say no! It is raised to the power of $x-1$. What?!!

I could take the log

$$(x-1)\log(3x-1)=0$$

$\log(3x-1)=0$

$3x-1=1$

$x=\frac{2}{3}$.

There is a problem, he told me there are more answers!!

I can't sees it. Can anyone please show me if there are more answer?

Answer 1

hint. If $$(x-1)\log(3x-1) = 0$$

Then either

$$\log(3x-1) = 0 $$

or

$$x-1 = 0 $$

Answer 2

Hint:$$(x-1)\log(3x-1)=0\implies \begin{array} xx-1=0\\ 3x-1=1\end{array}$$

Answer 3

You have it there!! $(x-1) \log(3x-1)= 0$ iff $x-1 =0$ or $\log(3x-1)=0$, so you are only missing $x-1=0 \rightarrow x =1$

Answer 4

The domain it's $3x-1>0$.

Now, we have two cases:

1. $3x-1=1$, which gives $x=\frac{2}{3};$

2. $x-1=0$, which gives $x=1$, for which the domain holds and we got the answer: $$\left\{\frac{2}{3},1\right\}$$