5.1 Exponential Functions

5.1 Exponential Functions

Exponential Function

• Definition: The function defined by $f(x) = b^x$ where $b > 0$ and $b \neq 1$ is called an exponential function with base $b$ and exponent $x$.

Laws of Exponential Functions

• Given that $a$ and $b$ are positive numbers and $x$ and $y$ are real numbers, the following properties hold:

  1. Product of Powers Property: $b^x \cdot b^y = b^{x+y}$

  2. Quotient of Powers Property: $\frac{b^x}{b^y} = b^{x-y}$

  3. Power of a Power Property: $(b^x)^y = b^{xy}$

  4. Power of a Product Property: $(ab)^x = a^x \cdot b^x$

  5. Power of a Fraction Property: $\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}$

  6. Comparison of Exponents: If $b^m = b^n$, then $m = n$.

Properties of the Exponential Function

• The exponential function $y = b^x$ with $b > 0$ and $b \neq 1$ has the following properties:

  1. Domain: The domain of the function is all real numbers, expressed as $(-\infty, \infty)$.

  2. Range: The range of the function is positive real numbers, expressed as $(0, \infty)$.

  3. Graph Intersection: The graph passes through the point $(0, 1)$, since $b^0 = 1$.

  4. Continuity: The exponential function is continuous on its entire domain $(-\infty, \infty)$.

  5. Behavior Analysis:

  • It is increasing on $(-\infty, \infty)$ if $b > 1$.

  • It is decreasing on $(-\infty, \infty)$ if $0 < b < 1$.

The Base e

• Value of e: $e \approx 2.718281828…$

• Limit Definition: $e$ can be defined as: $e = \lim_{m \to \infty} \left(1 + \frac{1}{m}\right)^m$.

• The natural exponential function is given by:

  • $y = e^x$

  • $y = e^{-x}$.

Examples

Example 1: Simplify the following expressions.

• (a) $31/4 \cdot 9^{-5/8}$

• (b) $\left(x^{-b/a}\right)^{-a/b}$

Example 2: Solve the following equations for $x$.

• (a) $e^{x-2} = e^{2x+1}$

• (b) $8^x = \left(\frac{1}{32}\right)^{x-2}$

Example 3: Sketch the graph of the given functions on the same axes.

• $y = 40.5^x$ and $y = 4^{-0.5^x}$

Supplemental Examples

Example 4: Simplify the following expressions.

• (a) $\frac{4b^{-4}}{12b^{-6}}$

• (b) $y^{-3/2}y^{5/3}$

• (c) $\left(\frac{-1}{2}\right)^2 \cdot (-3)$

Example 5: Function Evaluation

• A function $f$ has the form $f(x) = Ae^{kx}$.

• If it is known that $f(0) = 100$ and $f(1) = 120$, find $A$ and $k$.