Final answer:
To compare f(x) = log7x and g(x) = log4x, we examine when f=g, f>g, and fg, it is when x<1, and for f1.
Step-by-step explanation:
To compare the graphs of the logarithmic functions f(x) = log7x and g(x) = log4x, we need to understand the properties of logarithms. The function f(x) = log7x means that for every x, we are finding the power to which 7 must be raised to obtain that x value. Similarly, g(x) = log4x means finding the power to which 4 must be raised to obtain x.
When comparing the two functions, we can look for when f=g, when f>g, and when f<g. Since logarithmic functions are continuously increasing, the point at which f(x) equals g(x) occurs at x=1, because any logarithm of 1 is 0, regardless of the base. Additionally, this occurs at the intersection point when x equals the common base of the functions if there is one, but in this case, since 7 and 4 are different, there is no other common point where they will intersect.
The graph of f(x) will grow slower than g(x) because the base 7 is larger than base 4. This means that f(x) will be greater than g(x) for values of x less than 1, because within this interval, a smaller base grows faster. Conversely, for values of x greater than 1, g(x) will be larger than f(x), since for larger values of x, a larger base grows slower.