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Compare the graphs of the logarithmic functions f(x) = log,x and g(x) = log4x. For what values of x is f=g, f>g, and f you know.

User Nico Kaag
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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.

User Schystz
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