Exponential Parameters: the 'b' Value

1. Fundamental Concepts
Definition: Exponential functions have the form
$$f(x) = a \cdot b^x, \quad b > 0, \; b \neq 1.$$
Growth vs. Decay:
If $b > 1$ , the function shows exponential growth.
If $0 < b < 1$ , the function shows exponential decay.
Graph Description:
Growth ( $b > 1$ ): the curve rises sharply to the right.
Decay ( $0 < b < 1$ ): the curve falls toward zero as $x$ increases.

2. Key Ideas
The larger the value of $b > 1$ , the faster the growth.
The smaller the value of $0 < b < 1$ , the faster the decay.
The base $b$ directly determines the function’s shape.

3. Examples
Example 1 (Growth):
$$f(x) = 2^x \quad (b=2,\; \text{growth})$$
Example 2 (Decay):
$$g(x) = \left(\tfrac{1}{2}\right)^x \quad (b=\tfrac{1}{2},\; \text{decay})$$

4. Calculation Practice
Problem 1: Suppose $f(x) = b^x$ . If $f(3) = 27$ , find $b$ .
$$b^3 = 27 \quad \Rightarrow \quad b = 3$$
Problem 2: Suppose $g(x) = b^x$ . If $g(2) = \tfrac{1}{16}$ , find $b$ .
$$b^2 = \tfrac{1}{16} \quad \Rightarrow \quad b = \tfrac{1}{4}$$