How to simplify this formula: $ x^{\ln \left( 3 \right)}-3^{\ln \left( x \right)}$?

Question

Here's the formula: $$ x^{\ln \left( 3 \right)}-3^{\ln \left( x \right)}$$

I know it's equal to $0$ because I've tried different values for $x$, but how do I solve it, how do I simplify it to $0$?

Go here $\longrightarrow$ https://math.stackexchange.com/questions/320116/logarithm-proof-problem-a-log-b-c-c-log-b-a and substitue $b = e$. — Mr Pie, Mar 04 '18 at 00:04

score 7 · Accepted Answer · answered Mar 03 '18 at 23:58

7

First $$ x^{\ln(3)}=e^{\ln(3)\ln(x)} $$ And $$ 3^{\ln(x)}=e^{\ln(x)\ln(3)} $$

So yes it values $0$.

answered Mar 03 '18 at 23:58

Atmos

it's funny that I can solve harder formulas but the most simple I can not. Any tips on how to overcome that ? Practice or memorize? ;) – n00b Mar 04 '18 at 00:20
2

Practice is better IMO. – Oscar Lanzi Mar 04 '18 at 00:33

score 6 · Answer 2 · answered Mar 03 '18 at 23:59

6

Hint: For any positive real number $r$, $r=e^{\ln (r)}$. Apply this now to $r=x$ and $r=3$.

answered Mar 03 '18 at 23:59

Barry Cipra

1

Ah, I see three of us have posted almost simultaneously! – Barry Cipra Mar 04 '18 at 00:00
What is the problem? – n00b Mar 04 '18 at 00:14
@user32073, I'm afraid I don't understand your comment. – Barry Cipra Mar 04 '18 at 00:18
what is the problem about the posts being posted almost simultaneously? – n00b Mar 04 '18 at 00:19
@user32073, oh, there's no problem, I was just acknowledging that three of us had almost the exact same idea -- a case of great minds thinking alike.... – Barry Cipra Mar 04 '18 at 00:23

score 3 · Answer 3 · edited Mar 04 '18 at 00:00

3

Hint: use the fact that $a^x=e^{x\ln a}$ for $a>0$.

edited Mar 04 '18 at 00:00

Barry Cipra

answered Mar 03 '18 at 23:58

Tsemo Aristide

Mohammad Riazi-Kermani · Answer 4 · 2018-03-04T00:52:29.663

2

Note that for positive values of $x$,$$ x^{\ln \left( 3 \right)}=3^{\ln \left( x \right)}= e^{ln(3).ln(x)}$$

Therefore, $$ x^{\ln \left( 3 \right)}-3^{\ln \left( x \right)}=0$$

edited Mar 04 '18 at 00:52

answered Mar 04 '18 at 00:00

2

You are initially assuming the answer equals $0$. Why? – Mar 04 '18 at 00:01
@idk Because making an assumption is a good first step of proving statements (especially by contra-diction). For example, we may not know if $\sqrt 2$ is rational, or irrational. So, we assume $\sqrt 2 = p/q$ and where does this lead us? – Mr Pie Mar 04 '18 at 00:07
1

@user477343 What you are believing is incorrect. Here, you have to prove $a-b=0$, and the answer is assuming $a-b=0$ to prove $a-b=0$. In the case of $\sqrt 2$, you are assuming the OPPOSITE of what is correct to prove what is correct. – Mar 04 '18 at 00:30
@idk ahhh I see... thank you for letting me know :) – Mr Pie Mar 04 '18 at 01:09

4 Answers4