Final answer:
The exponential function y=2^x and the logarithmic function y=log_2x are inverses of each other, resulting in their graphs being symmetric over the line y=x. The correct statements are that these functions are inverses, and their graphs have symmetry over the line y=x, not y=0.
Step-by-step explanation:
The relationship between y=2^x and y=log_2x can be understood through various properties of exponents and logarithms. To tackle the given statements, let's examine each one:
- The graphs of the functions are symmetric about the line y=0. This statement is incorrect; the symmetry for inverses is over the line y=x, not y=0.
- The equation y=log_2x is the logarithmic form of y=2^x. This statement is correct; it represents the same relationship in different forms.
- The functions are inverses of each other. This statement is correct; the exponential and logarithmic functions are known to be inverses of each other.
- The graphs of the functions are symmetric to each other over the line y=x. This statement is correct; by definition, the graphs of inverse functions are symmetric over the line y=x.
To summarize, the functions y=2^x and y=log_2x are inverses, meaning one 'undoes' the other. This inverse relationship does result in their graphs being symmetric over the line y=x, but not y=0. The logarithmic function y=log_2x is simply another way to express the relationship embedded in the exponential form y=2^x.
Mention of the correct option in the final answer: The statements that express the correct relationship between y=2^x and y=log_2x are that the functions are inverses of each other, and their graphs are symmetric over the line y=x.