How do you solve this logarithm?

Question

Solve for $x$ in the following:

$$x = 9^{\log_{3}\left(2\right)}$$

The answer is $4$, but why?

Answer 1

Well, $9=3^2$. So what we really have is $$ x=9^{\log_3 2}=\left( 3^2 \right)^{\log_3 2}=3^{2\log_3 2}=3^{\log_3 2^2} $$ But then that is $$ x=3^{\log_3 2^2}=3^{\log_3 4}=4 $$

Answer 2

$$9^{\log_32}=({3^2})^{\log_32}=3^{2(\log_32)}$$

now use the logarithmic identity $x\log_nb=log_nb^x$ to get:

$$3^{2(\log_32)}=3^{(\log_32^2)}=3^{(\log_34)}=4$$

Answer 3

$$9 = 3^2, \space a \log x = \log x^a, \space a^{\log_a x} =x$$

$$x = 9^{\log_3 2} = 3^{2 \log_3 2} = 3^{\log_3 2^2} = 3^{\log_3 4}$$

$$\therefore 3^{\log_3 4} = 4$$

Answer 4

$\log_3 x=\log_3(9^{\log_3 2})=\log_3 2\cdot\log_3 9=2\log_3 2=\log_3 4$