A business owner uses the exponential function $f(x) = a^x$ to track the value of a piece of equipment over time. According to the mathematical definition of an exponential function, why must the base $a$ be strictly positive ($a > 0$)?

A business analyst uses an exponential function $f(t) = a^t$ to project a company's revenue growth over time. True or False: In the mathematical definition of an exponential function, the base $a$ must be strictly positive ($a > 0$) to ensure the function is defined for all real values of $t$.

Requirement for a Positive Base in Exponential Functions

A supply chain analyst is using an exponential growth model $f(x) = a^x$ to predict the increase in regional shipping demands. To ensure that the model is mathematically valid and produces real-number results for all possible real values of $x$, the analyst must recall that the base $a$ is required to be strictly ____.

A scientist is using a growth model of the form $f(x) = a^x$ to track the spread of a biological culture. In this model, the base $a$ is constrained by specific mathematical rules. Match each part of the model's definition with its correct requirement or rationale.

A business analyst at a logistics firm is designing a spreadsheet model to project shipping volumes using an exponential growth function of the form $f(x) = a^x$. During a system test, the analyst wonders why the software's formula validator automatically rejects any base $a$ that is not strictly positive ($a > 0$). To understand the mathematical rationale behind this rule, arrange the following steps in the correct logical order to demonstrate why a negative base fails to produce real-number outputs.