Final answer:
Logarithmic functions f(x) = log x and g(x) = log 4x never intersect as log 4 is always positive, thus g(x) is always greater than f(x) for all x > 0. The logarithmic property log a - log b = log(a/b) helps understand this relationship, and the increasing nature of a logarithm on a logarithmic scale illustrates it.
Step-by-step explanation:
When comparing the graphs of logarithmic functions f(x) = log x and g(x) = log 4x, a few properties of logarithms are pivotal for understanding the relationship between these two functions. By using the logarithm property, log a - log b = log(a/b), we can understand the relationship between these two functions and determine where f(x) = g(x), f(x) > g(x), and f(x) < g(x).
First, to find when f(x) equals g(x), we set log x equal to log 4x. This equation simplifies to log x = log x + log 4, leading to the conclusion that this equation holds true only if log 4 equals zero, which is not the case. Therefore, f(x) and g(x) are never equal.
Since log 4 is positive, for any value of x, log 4x is always greater than log x, meaning that g(x) > f(x) for all x > 0. A logarithmic plot further illustrates this, showing the graph of g(x) being a vertical shift of the graph of f(x) upwards by log 4 units, given that the logarithmic function is always increasing as x increases.
To clarify, this increase is not linear; in a logarithmic plot, a constant increase in x leads to a progressively smaller increase in the log value of x—a concept fundamental to the utility of logarithmic scales. This behavior is evident in phenomena like equilibrium calculations and exponential growth.