Final answer:
The question involves transformations of the exponential function f(x) = 3^x, including cubing the function, evaluating it at a certain point, and understanding properties of logarithms and exponential functions. Dimensional consistency and power series are also alluded to.
Step-by-step explanation:
The subject of the question involves mathematical transformations, specifically involving the function f(x) = 3^x. When dealing with transformations of exponential functions such as cubing, it is important to understand the difference between cubing the base number and affecting the exponent. To cube an exponential function, you simply raise the base to the power of three and multiply the original exponent by three, as seen in the provided example where (53)4 becomes 512.
When considering functions at a given point, such as x = 3, it is necessary to evaluate both the value and the slope (derivative) of the function at that point to determine which option corresponds to the conditions provided.
In the context of logarithmic and exponential functions, it is critical to remember fundamental properties such as the fact that logarithms of numbers raised to exponents are the product of the exponent and the logarithm of the number. Additionally, exponential and natural logarithm functions (e^x and ln(x)) are inverse functions of each other, hence they 'undo' each other.
Understanding the concept of dimensional consistency is vital when working with functions in calculus, ensuring that you cannot combine unlike terms, such as adding apples and oranges, which would be dimensionally inconsistent. Functions like trigonometric functions, logarithms, and exponential functions can be represented as infinite sums when expanded via power series, which conveys that they are mathematically well-defined beyond just integer exponents.