The function $y = \log_4 x$ represents a logarithmic function with base 4. The characteristics of this function include:

1. Domain: $x > 0$, since the logarithmic function is only defined for positive values of $x$.
2. Range: All real numbers, since $y$ can take any value from $-\infty$ to $+\infty$.
3. X-intercept: This function intersects the x-axis at $x = 1$ because $\log_4(1) = 0$.
4. Asymptote: The function has a vertical asymptote at $x = 0$.

For $y = \log_4 x$, the graph:

• Passes through the point $(1, 0)$.
• Rises as $x$ increases and approaches infinity.
• Approaches negative infinity as $x$ approaches 0 from the right.

Given these characteristics, Option C is the correct choice. This option shows:

• The curve rising steeply from the x-axis near $x = 0$ and gradually increasing as $x$ increases.
• The curve passing through the x-axis at $x = 1$, where $y = 0$.

Would you like more details on interpreting logarithmic function graphs?

Here are five questions for further exploration:

1. How does changing the base of the logarithmic function affect its graph?
2. What happens to the intercept if the function changes to $y = \log_4(x - c)$?
3. How do you find the asymptote for transformed logarithmic functions?
4. What is the effect of a vertical shift on the graph of $y = \log_4 x$?
5. How does the graph of $y = \log_4 x$ compare to $y = \ln x$?

Tip: When graphing logarithmic functions, always remember the x-intercept occurs where $x = 1$ and identify the vertical asymptote at $x = 0$ for $\log_b x$ functions.