Final answer:
The graphs of the equations y = log5 x and y = 5^x are not symmetric over the line y = 0, and y = 5^x is not the exponential form of y = log5 x, it should be x = 5^y. However, these functions are indeed inverses of one another and their graphs are symmetric over the line y = x.
Step-by-step explanation:
The statements concerning the equations y = log5 x and y = 5x require an understanding of logarithmic and exponential functions.
- a) False. The graphs of these equations are not symmetric over the line y = 0. The graph of y = log5 x is a logarithmic curve that increases steadily to the right and approaches negative infinity as x approaches zero from the right. Whereas y = 5x is an exponential graph that shoots up quickly as x increases. Symmetry over the line y = 0 would imply that the graphs are mirror images across the horizontal axis, which is not the case here.
- b) False. The correct exponential form of the equation y = log5 x would be x = 5y. The logarithmic function represents the exponent to which the base (in this case, 5) must be raised to yield x.
- c) False. The functions are indeed inverses of each other. For logarithmic functions and their corresponding exponential functions, one 'undoes' the effect of the other. This inverse relationship means that logb x and by are inverse functions if y = logb x.
- d) True. The equations are symmetric over the line y = x. The inverse relationship between logarithmic and exponential functions is visually represented by symmetry about the line y = x. If one graph is reflected over this line, it would lie directly on top of the other graph.