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

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 $0$?

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

So yes it values $0$.

• 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 4, 2018 at 0:20
• Practice is better IMO. Mar 4, 2018 at 0:33

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

• Ah, I see three of us have posted almost simultaneously! Mar 4, 2018 at 0:00
• What is the problem?
– n00b
Mar 4, 2018 at 0:14
• @user32073, I'm afraid I don't understand your comment. Mar 4, 2018 at 0:18
• what is the problem about the posts being posted almost simultaneously?
– n00b
Mar 4, 2018 at 0:19
• @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.... Mar 4, 2018 at 0:23

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

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

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

• You are initially assuming the answer equals $0$. Why?
– user535339
Mar 4, 2018 at 0:01
• @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? Mar 4, 2018 at 0:07
• @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.
– user535339
Mar 4, 2018 at 0:30
• @idk ahhh I see... thank you for letting me know :) Mar 4, 2018 at 1:09