Logarithms simplification struggle

Question

Question : Suppose that $x$ is a positive real number and $y = \log_9 x$. Express $3^y$ as function of $x$ in as simple a form as possible.

What I am planning to do :

So $3^y = \log_9(x^3)$

Am I right?

Answer 1

So

$$9^y = 9^{\log_9 x}=x$$ which means $$(3^2)^y = x$$ which means $$(3^y)^2 = x$$ or which means $$3^y = \sqrt{x}$$

Answer 2

Hint: Prove that $$3^{\log_{9}{x}}=\sqrt{x}$$ Second hint: $$\frac{\ln(x)\ln(3)}{2\ln(3)}=\frac{1}{2}\ln(x)$$

Answer 3

To start the problem, we want to find $3^y$ as a function of $x$. So, substitute $y=\log_9 x$ into the expression $3^y$. Then our function is $3^{\log_9 x}$. Note that function is identical to our final answer but not yet completely simplified. So, a neat way of checking that your final answer is correct could be to see if $3^{\log_9 x}$ gives you the same value as your simplification. This check shows that $\log_9(x^3)$ was incorrect.

But how can we simplify $3^{\log_9 x}$? Let's give this expression a name, to make the algebra easier. For instance, $u=3^{\log_9 x}$. So then how can we use the laws of logarithms to simplify $u$?

Hint 1: $\quad\log_b(a^n)=n\log_b(a)$

Hint 2: $\quad3=9^{1/2}$

These laws give a methodical approach to solving the problem and should work as long as you're consistent and keep track of your bases. The other two answers posted can lead you to the right answer but skip a few steps in doing so.