Integrals, Exponential Functions, and Logarithms

Learning Objectives

After completing this study guide, you should be able to:

Write the definition of the natural logarithm as an integral.
Recognize the derivative of the natural logarithm.
Integrate functions involving the natural logarithmic function.
Define the number $e$ through an integral.
Recognize the derivative and integral of the exponential function.
Prove properties of logarithms and exponential functions using integrals.
Express general logarithmic and exponential functions in terms of natural logarithms and exponentials.

The "Big Idea"

Prior to integral calculus, exponential functions like $2^x$ were only rigorously understood when $x$ was a rational number (like $2^{3/2}$ ). What does $2^{\pi}$ actually mean?

Calculus resolves this by working backward. Instead of defining exponents first, we define the natural logarithm as the accumulated area under the curve of $y = \frac{1}{t}$ . From this rock-solid geometric foundation, we define its inverse function, $e^x$ . Finally, we use $e^x$ to build a rigorous definition for any exponential base: $a^x = e^{x \ln a}$ . This brilliant workaround bridges the gap between geometry and algebra, allowing us to perform calculus on general exponential and logarithmic functions.

Key Terms & Glossary

Natural Logarithm ( $\ln x$ ): Defined rigorously as the definite integral of $\frac{1}{t}$ from 1 to $x$ .
The Number $e$ : The unique real number such that the area under $\frac{1}{t}$ from 1 to $e$ is exactly 1 (i.e., $\ln(e) = 1$ ).
General Exponential Function ( $a^x$ ): Defined for any base $a > 0$ and any real number $x$ using the natural exponential: $a^x = e^{x \ln a}$ .
General Logarithm ( $\log_a x$ ): The inverse of the general exponential function, expressible via the change-of-base formula.

Hierarchical Outline

1. Foundation: The Natural Logarithm
- The Power Rule Failure: Why $\int x^{-1} dx$ requires a new function.
- Integral Definition: $\ln x$ as net area.
2. The Natural Exponential Function
- Defining $e$ : The specific x-value where area equals 1.
- Inverse Relationship: $e^x$ defined as the inverse of $\ln x$ .
3. Generalizing Bases
- Irrational Exponents: Rigorously defining $a^x$ via $e^{x \ln a}$ .
- Calculus of $a^x$ : Derivatives and integrals incorporating $\ln a$ .
- General Logarithms: $\log_a x$ as the inverse of $a^x$ .

Formula / Concept Box

Function Type	Definition	Derivative	Integral
Natural Log	$\ln x = \int_1^x \frac{1}{t} dt$	$\frac{d}{dx}(\ln x) = \frac{1}{x}$	$\int \ln x dx = x \ln x - x + C$
Natural Exp	$y = e^x \iff \ln y = x$	$\frac{d}{dx}(e^x) = e^x$	$\int e^x dx = e^x + C$
General Exp	$a^x = e^{x \ln a}$	$\frac{d}{dx}(a^x) = a^x \ln a$	$\int a^x dx = \frac{a^x}{\ln a} + C$
General Log	$\log_a x = \frac{\ln x}{\ln a}$	$\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$	N/A (usually convert to $\ln$ )

[!IMPORTANT] When differentiating or integrating general exponential functions ( $a^x$ ), you will always multiply or divide by the constant $\ln a$ .

Visual Anchors

1. The Logical Progression of Exponential Definitions

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Figure 1 — Mermaid diagram

2. Geometric Definition of the Natural Logarithm

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Figure 2 — TikZ diagram

Definition-Example Pairs

General Exponential Function ( $a^x$ )
- Definition: A function of the form $f(x) = a^x$ , mathematically defined as $e^{x \ln a}$ to allow for continuous domains.
- Real-world Example: Calculating the future value of an investment compounding annually at 5% for an irregular time period like $\pi$ years ($1.$05^\pi$$).
General Logarithm ( $\log_a x$ )
- Definition: The inverse of $a^x$ , written in terms of the natural logarithm as $\frac{\ln x}{\ln a}$ .
- Real-world Example: Determining the exact magnitude of an earthquake on the Richter scale, which is fundamentally a base-10 logarithmic scale.

Comparison Tables

Base $e$ vs. Base $a$ Calculus

Operation	Natural Base ( $e$ )	General Base ( $a > 0,$ a \neq 1$$)	Connection
Function	$f(x) = e^x$	$f(x) = a^x$	$a^x = e^{$ x \ln a $}$
Derivative	$\frac{d}{dx}(e^x) = e^x$	$\frac{d}{dx}(a^x) = a^x \ln a$	Chain rule on $e^{$ x \ln a $}$
Integral	$$\int e^x $dx = e^x + C$	$\int a^x$ dx $= \frac{a^x}{\ln a} + C$	U-substitution on $$\int e^{x \ln a} $dx$

Worked Examples

Example 1: Differentiating a General Exponential

Problem: Find $\frac{d}{dx}(5^{x^2})$ .

▶Click to view step-by-step solution

Identify the rule: The derivative of $a^u$ is $a^u \ln(a) \cdot \frac{du}{dx}$ .
Set up parts:
- $a = 5$
- $u = x^2$
- $\frac{du}{dx} = 2x$
Substitute into formula: $\frac{d}{dx}(5^{x^2}) = 5^{x^2} \cdot \ln(5) \cdot 2x$
Simplify: $2x \cdot 5^{x^2} \ln 5$

Example 2: Integrating a General Exponential (Text Example 2.39)

Problem: Evaluate $$\int 3^x $dx$ .