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 $0$?

Answer 1

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

So yes it values $0$.

Answer 2

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

Answer 3

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

Answer 4

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

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