compare the graphs of the logarithmic functions f(x)=log7x and g(x)=log4x. for what values of x is f=g, and f>g, and f<g. explain how you know.
To compare the graphs of the logarithmic functions f(x) = log7x and g(x) = log4x, we first need to understand the properties of logarithmic functions.
1. Same base with different coefficients:
- The coefficient in front of the logarithm affects the vertical shift of the graph, but it does not change the shape.
- It results in a vertical stretch or compression of the graph.
2. Different bases:
- Different bases affect the horizontal stretch or compression of the graph.
- They also determine the vertical shift of the graph.
Now, let's analyze the specific functions f(x) = log7x and g(x) = log4x:
1. Horizontal Shift:
- Both functions have a horizontal shift of 0 since there are no transformations applied to the logarithm's argument (x).
2. Vertical Shift:
- f(x) = log7x: This function has a vertical shift of 0.
- g(x) = log4x: This function also has a vertical shift of 0.
3. Base Comparison:
- f(x) = log7x: This function has a base of 7.
- g(x) = log4x: This function has a base of 4.
Knowing these properties, we can now compare the graphs:
1. f(x) = g(x) (f = g):
To find x values where f(x) = g(x), we set the functions equal to each other:
log7x = log4x
Since both logarithms have the same base, we can equate the arguments:
7x = 4x
3x = 0
x = 0
f(x) = g(x) when x = 0.
2. f(x) > g(x):
For f(x) > g(x), we need to compare the two functions to see when one is greater than the other.
- Since log7x has a greater base than log4x, log7x will increase faster than log4x for all positive x values.
- Hence, f(x) > g(x) for all positive x values.
3. f(x) < g(x):
For f(x) < g(x), we need to compare the two functions to see when one is less than the other.
- Since log4x has a smaller base than log7x, log4x will increase slower than log7x for all positive x values.
- Hence, f(x) < g(x) for all positive x values.
In summary:
- f(x) = g(x) when x = 0.
- f(x) > g(x) for all positive x values.
- f(x) < g(x) for all positive x values.