Final answer:
The correct equation for an exponential function with a base greater than zero is a) y = aⁿ, where a > 0. This is the standard form and fits the requirement specified in the question.
Step-by-step explanation:
The student has asked to write an equation that defines the exponential function with a base greater than zero. An exponential function is generally written as y = aⁿ, where 'a' is the base of the exponential function and must be greater than zero, and 'n' is the exponent. To ensure that our exponential function has a base greater than zero, we can examine the options provided.
- a) y = aⁿ, where a > 0: This is the standard form of an exponential function, and it specifies that the base 'a' is greater than zero, which is required for our definition. This option fits the criteria perfectly.
- b) y = eⁿ, where e is the base of natural logarithms: This is also a correct description of an exponential function. Here, 'e' represents the mathematical constant approximately equal to 2.7183, which is indeed greater than zero.
- c) y = 2ⁿ, where 2 > 0: This option specifies that the base of the exponential function is 2, which is also greater than zero and thus defines a valid exponential function.
- d) y = (-3)ⁿ, where -3 > 0: This option incorrectly states that -3 is greater than zero. Since -3 is not greater than zero, it cannot be the base of an exponential function in the sense we want. Therefore, this option is incorrect.
By applying these characteristics and considering the need for the base to be above zero, the correct option is a) y = aⁿ, where a > 0. This is the most generalized form of an exponential function with the base a that is greater than zero, which can be expressed in terms of 'e' using the relationship ‘bⁿ = eⁿ ln b’ or the base-10 equivalent for various applications.