Compare the graphs of the logarithmic functions f left parenthesis x right parenthesis equals log subscript 7 baseline x and g left parenthesis x right parenthesis equals log subscript 4 baseline x. For what values of x is f = g, f greater than g, and f less than g? Explain how you know.
To compare the graphs of the logarithmic functions f(x) = log₇(x) and g(x) = log₄(x), we can examine their behavior and key properties:
1. Intercepts:
- For f(x) = log₇(x), the graph intersects the x-axis at x = 1 (since log₇(1) = 0), and it has no y-axis intercept.
- For g(x) = log₄(x), the graph does not intersect the x-axis or y-axis.
2. Asymptotes:
- Both functions f(x) and g(x) have vertical asymptotes. For f(x) = log₇(x), the vertical asymptote occurs at x = 0. For g(x) = log₄(x), the vertical asymptote occurs at x = 0 as well.
3. Shape:
- Logarithmic functions have an increasing but decreasing rate of change. This means that as x approaches infinity, the values of y increase more slowly. Similarly, as x approaches negative infinity, the values of y decrease more slowly.
Comparing f(x) = log₇(x) and g(x) = log₄(x):
- f(x) > g(x) when x > 1: For x values greater than 1, f(x) has a slower rate of increase than g(x). This can be seen by comparing the slopes of the logarithmic curves. For example, f(2) ≈ 0.431 and g(2) ≈ 0.602, f(3) ≈ 0.616 and g(3) ≈ 0.792, etc.
- f(x) < g(x) when 0 < x < 1: For x values between 0 and 1, f(x) has a steeper rate of decrease than g(x). This can be seen by comparing the slopes of the logarithmic curves. For example, f(0.5) ≈ -0.214 and g(0.5) ≈ -0.321, f(0.2) ≈ -0.435 and g(0.2) ≈ -0.602, etc.
- f(x) = g(x) when x = 1: Both f(x) and g(x) equal zero at x = 1, so the graphs coincide at this point.
In summary:
- f(x) > g(x) when x > 1.
- f(x) < g(x) when 0 < x < 1.
- f(x) = g(x) when x = 1.
These conclusions can be drawn by considering the general properties and behaviors of logarithmic functions, as well as by comparing specific values of f(x) and g(x) at different x points.