Final answer:
The transformation of the parent function f(x)=2^x represents exponential growth by doubling the output value at each step or interval. Expressing base-2 as e^(ln 2) links exponential functions to their inverse logarithmic functions, providing a mathematical trick to simplify expressions and calculations in exponential growth.
Step-by-step explanation:
The transformation of the parent function f(x)=2^x involves applying mathematical operations that alter its shape or position on the graph. A key aspect of understanding these transformations is recognizing the relationship between exponential functions and their inverses, particularly the natural logarithm (ln).
Using properties of exponents and logarithms, we can express any exponential growth as a function with the base e (Euler's number, approximately 2.7183). This is done by equating the base b to e raised to the power of the natural logarithm of b: b = e^(ln b). Specifically, for the base 2, we have 2 = e^(ln 2).
When considering the transformation of f(x)=2^x, we can view it in terms of exponential growth. At each step, or time interval, you multiply by 2. After n intervals, you've multiplied by 2 n times, resulting in 2^n. In the context of our function, this means for any value of x, the function multiplies the initial value (when x = 0) by 2 x times. This results in exponential growth, where the function's value doubles every time x is incremented by 1.
An understanding of these concepts is essential for comprehending how exponential functions can be transformed into functions using the base e, which can be more convenient for certain mathematical applications, due to the unique properties of the number e.