Graph the logarithmic function. Find any intercepts.
yequalslog Subscript 4 Baseline x
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Part 1
Choose the correct graph.
A.
-5
5
-5
5
x
y



A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 5 to 5 in increments of 1. From left to right, a solid curve falls steeply in quadrant 1 crosses the x-axis at 1, and then falls shallowly in quadrant 4. As x approaches 0 from the right, the curve approaches the positive y-axis from the right.
B.
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-5
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y



A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 5 to 5 in increments of 1. From left to right, a solid curve falls steeply in quadrant 2, crosses the y-axis at 1, and then falls shallowly in quadrant 1. As x increases towards positive infinity, the curve approaches the positive x-axis from above.
C.
-5
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-5
5
x
y



A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 5 to 5 in increments of 1. From left to right, a solid curve rises steeply in quadrant 4 crosses the x-axis at 1, and then rises shallowly in quadrant 1. As x approaches 0 from the right, the curve approaches the negative y-axis from the right.
D.
-5
5
-5
5
x
y




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.

Graph the logarithmic function. Find any intercepts. y equals log base 4 of x.

Learn how to graph the logarithmic function y = log_4 x and find its intercepts. This solution explores the function's domain, range, x-intercept, and asymptote, providing detailed insights for students in Grades 9-12.

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