Why does this problem have one less root for x?

Question

Problem statement (Problem $6$):

This is my work below:

The answer of $x$ is $2$ and $8$. but my answer is $1$/ $\sqrt{3}$ and it is a credible answer if you solve it. Why is this happening?

Your work yielded the answer $x = 81$. Are you mixing up the problems? — quasi, Feb 21 '18 at 07:24
The $x=2$ corresponds to $x-1=1$ and $1^a=1^b$ for every $a, b$ — kingW3, Feb 21 '18 at 07:30
Dude you did it so easily and I was like what the hell’s wrong with me since 2-3 days. Thanks though you could’ve written it as an answer you know...to get the upvote — Dilin Finn, Feb 21 '18 at 07:34
It's a hassle to write answers from the phone and imo 10k is enough for me, anyway it's mostly experience, the more problems you do the easier it will be for you to spot things like this. — kingW3, Feb 21 '18 at 07:44

score 3 · Answer 1 · answered Feb 21 '18 at 07:33

3

There are two answers, $81$ and $2$.

If $a$ is positive, and $s,t$ are real, the equation $$a^s = a^t$$ holds if and only if $a=1$ or $s=t$.

You analyzed the case $s=t$, which yields $x=81$.

You missed the case $a=1$, which yields $x=2$.

answered Feb 21 '18 at 07:33

quasi

score 1 · Answer 2 · answered Feb 21 '18 at 07:20

1

When $x=1/\sqrt3$, $x-1<0$ and so $(x-1)^7<0$. But $|x-1|>0$ and so $|x-1|^{\textrm{nasty expression}}>0$. Thus $1/\sqrt3$ cannot be a solution.

answered Feb 21 '18 at 07:20

Angina Seng

Yeah that’s resolved even if I put it in the nasty expression but how does x=2 come? I mean this question has repeated twice in the book and has 2 as an answer both times... – Dilin Finn Feb 21 '18 at 07:27

user · Answer 3 · 2018-02-21T07:36:55.273

1

Remember that you are looking for a solution under the condition $(x-1)>0$.

Note that since $t=\log_3 x$ the solutions are $x=81$ and $x=\frac1{\sqrt3}$.

As pointed out by quasi, the other solution come from the trivial case $x-1=1$.

edited Feb 21 '18 at 07:36

answered Feb 21 '18 at 07:23

user

Yeah but how does value of x come to be 2? I get the x-1>0 – Dilin Finn Feb 21 '18 at 07:27
That could be... the study material’s the nastiest then. This question was repeated further,in a multiple choice question. The three values we have -2,81,1/3^-1/2 were in the options and bam,2 was the answer again. Legend of Typos – Dilin Finn Feb 21 '18 at 07:32
A comment just answered that... – Dilin Finn Feb 21 '18 at 07:33
@DilinFinn Yes of course, we have also the case $x-1=1$ which leads to x=2. – user Feb 21 '18 at 07:36

3 Answers3