The problem asks to rewrite the logarithmic function $y = \log_3(x)$ using the change-of-base formula and base $e$ logarithms (natural logarithms).

1. Problem Description

The problem asks to rewrite the logarithmic function $y = \log_3(x)$ using the change-of-base formula and base $e$ logarithms (natural logarithms).

2. Solution Steps

The change-of-base formula states that for any positive numbers $a$, $b$, and $x$ where $a \neq 1$ and $b \neq 1$,

$$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$$

In our case, we want to rewrite $y = \log_3(x)$ using base $e$ (natural logarithms). So, $a = 3$, $x = x$, and $b = e$. Therefore, we have:

$$\log_3(x) = \frac{\log_e(x)}{\log_e(3)}$$

Since $\log_e(x)$ is the natural logarithm of $x$, we can write it as $\ln(x)$. Therefore,

$$\log_3(x) = \frac{\ln(x)}{\ln(3)}$$

So, the expression for $y$ becomes:

$$y = \frac{\ln(x)}{\ln(3)}$$

3. Final Answer

$y = \frac{\ln(x)}{\ln(3)}$