Exponential Functions 

Exponential Functions Definition

An exponential function is a mathematical function in the form:

$f(x)=a\cdot b^x$ 

where:

• a ≠ 0 (initial value),
• b > 0 and b ≠ 1 (base of the exponential),
• x is the exponent (independent variable).

The most common base in calculus is the constant $e\approx 2.718$, leading to the natural exponential function:

$f(x)=e^x$

1.0What is the Exponential Function?

In simpler terms, an exponential function grows (or decays) rapidly because the variable is in the exponent. These functions model population growth, radioactive decay, compound interest, and many natural processes. 

2.0Exponential Function Formulas

1. Basic Exponential Function

$f(x)=a\cdot b^x$

2. Natural Exponential Function

$f(x)=e^x$

3. Compound Interest Formula (Exponential Growth)

$A=P{\left(1+\frac{r}{n}\right)}^{nt}$

4. Continuous Growth/Decay

$A=P{e}^{rt}$

3.0Graphing Exponential Functions

Characteristics of the exponential functions graph:

• Always passes through the point (0, a)
• Rapid growth for b > 1
• Decay for 0 < b < 1
• Never touches the x-axis — horizontal asymptote at y = 0
• Always positive for real values of x

Example:

Graph of $f(x)=2^x$

• Increases as x increases
• Horizontal asymptote at  y = 0

$Graphoff(x)=(1/2{)}^x$

• Decreases as x increases (exponential decay)

4.0Exponential Functions Asymptotes

The horizontal asymptote of an exponential function is typically:

y = 0 

Unless the function is transformed, e.g. $f(x)=2^x+3$, then the asymptote is y = 3.

5.0How to Calculate Exponential?

To calculate exponential values:

• Use calculators or logarithm tables
• For $e^x$, scientific calculators have a dedicated EXP or $e^x$ button
• For base b, use:

$b^x=e^{x\ln b}$

6.0Exponential Properties

The following laws govern exponential expressions:

1. $b^x\cdot b^y=b^{x+y}$
2. $\frac{b^x}{b^y}=b^{x-y}$
3. ${\left(b^x\right)}^y=b^{xy}$
4. $b^{-x}=\frac{1}{b^x}$
5. $b^0=1$

These exponential properties simplify calculations and algebraic expressions.

7.0Logarithmic and Exponential Functions

Logarithmic functions are inverses of exponential functions.

If:

$y=b^x\Rightarrow {\log }_b(y)=x$

This connection is crucial for solving exponential equations:

$2^x=8\Rightarrow x={\log }_2(8)=3$

8.0Exponential Functions Examples

Example 1:Evaluate $f(x)=2^{x}atx=3$

Solution: 

$f(3)=2^3=8$

Example 2: Find the value of $e^2$ (rounded)

Solution: 

$e^2\approx 7.389$

Example 3: If a population doubles every 3 years, starting at 500, find the population after 9 years.

Solution: 

$P=500\cdot 2^{9/3}=500\cdot 2^3=500\cdot 8=4000$

Example 4: Solve: $e^{2x}-5e^x+6=0$

Solution:
Let $t=e^x$, then t > 0:

$t^2-5t+6=0(t-2)(t-3)=0\Rightarrow t=2\text{ or }3$

Back-substitute:

$\begin{array}{rl}& e^x=2\Rightarrow x=\ln 2\\ & e^x=3\Rightarrow x=\ln 3\end{array}$

Answer: x = ln 2, ln 3

Example 5: Evaluate: ${\lim }_{x\to 0}\frac{e^{2x}-1}{x}$

Solution:
Using limit property:

$\begin{array}{rl}& \underset{x\to 0}{\lim }\frac{e^{kx}-1}{x}=k\\ & \text{ Thus, }\\ & \underset{x\to 0}{\lim }\frac{e^{2x}-1}{x}=2\end{array}$

Answer: 2

Example 6: Evaluate: ${\int }_0^1{e}^{-x^2}dx$

Solution:
This is a Gaussian integral, no elementary form exists.

Approximation or error function:

${\int }_0^1{e}^{-x^2}dx\approx 0.7468\text{ (from standard tables) }$

Answer: Approximate value ≈ 0.7468 

Example 7: Find: $\frac{d}{dx}\left(e^{\sin x}\right)$

Solution:
Chain rule:

$e^{\sin x}\cdot \cos x$

Answer: $e^{\sin x}\cdot \cos x$

Example 8: Solve: $\ln \left(e^x-2\right)=1$

Solution:
Take exponential on both sides:

$\begin{array}{rl}& e^x-2=e^1\\ & \Rightarrow e^x=e+2\\ & \Rightarrow x=\ln (e+2)\end{array}$

Answer: $x=\ln (e+2)$

Example 9: Find domain: $f(x)=\sqrt{e^x-4}$ 

Solution:

$\begin{array}{rl}& e^x-4\ge 0\\ & \Rightarrow e^x\ge 4\\ & \Rightarrow x\ge \ln 4\end{array}$

Answer: $x\ge \ln 4$

Example 10: Solve: $\frac{dy}{dx}=y\cdot e^x$

Solution:
Separate variables:

$\begin{array}{rl}& \frac{dy}{y}=e^{x}dx\\ & \Rightarrow \ln y=e^x+C\\ & \Rightarrow y=A{e}^{e^x}\end{array}$

Answer: $y=A{e}^{e^x}$, where A is constant.

9.0Exponential Functions Differentiation

Let’s find the derivative of $f(x)=a\cdot e^{bx}$

Derivative:

$\frac{d}{dx}\left(a\cdot e^{bx}\right)=a\cdot b\cdot e^{bx}$

Example:

$f(x)=3e^{2x}\Rightarrow f^{\prime }(x)=6e^{2x}$

For f(x)=b^x, use:

$\frac{d}{dx}\left(b^x\right)=b^x\ln b$

4. What is the exponential function?

Ans: It’s a function where the variable is in the exponent, usually of the form $f(x)=a\cdot b^x$, used to model growth and decay.

5. How do you calculate exponential values?

Ans: Use formulas or a calculator. Use $e^x$ for natural exponentials or convert using $b^x=e^{x\ln b}$ .

10.0Practice Questions on Exponential Functions

1. Differentiate: $f(x)=5e^{3x}$
2. Evaluate: $3^4$
3. Solve for x: $2^x=16$
4. Find the horizontal asymptote of ​​$f(x)=2^x-4$
5. If A=1000 $e^{0.05t}$, find A when t = 10