Unexplained log simplification, can someone show how?

Question

An online course I am following performs the following $\log$ simplification without explanation:

$$ 9^{\log_3n} = n^2 $$

I am sure this is somewhat simple, but I just cannot see the simplification using the properties of log. I would greatly appreciate the help.

score 7 · Answer 1 · answered Oct 10 '20 at 20:38

7

$\displaystyle 9^{\log_3 n} = 3^{2\log_3 n} = (3^{\log_3 n})^2 = n^2$

answered Oct 10 '20 at 20:38

gandalf61

Can you explain the last step? I am not seeing how the term within the parentheses gets simplified to n. – Mr.Mips Oct 10 '20 at 20:48
4

@Mr.Mips That follows straight from the definition of $\log_3$. – gandalf61 Oct 10 '20 at 20:59

score 3 · Accepted Answer · answered Oct 10 '20 at 20:49

3

By definition

$$y=\log_3n \iff 3^y =n$$

then

$$9^{log_3n} = 9^y=(3\cdot 3)^y=3^y\cdot 3^y=n\cdot n=n^2$$

answered Oct 10 '20 at 20:49

user

2 Answers2