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

[!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

2. Geometric Definition of the Natural Logarithm

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

Worked Examples

Example 1: Differentiating a General Exponential

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

▶Click to view step-by-step solution

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

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

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

▶Click to view step-by-step solution

1. Identify the rule: From Theorem 2.20, \int a^x$ dx $= \frac{a^x}{\ln a} + C.
2. Identify the base: Here, $a = 3$.
3. Apply the formula directly: $\int 3^x dx = \frac{3^x}{\ln 3} + C$

Checkpoint Questions

1. Why can't we use the power rule \int x^n$ dx $= \frac{x^{n+1}}{n+1} to find the antiderivative of $x^{-1}$?
2. What is the fundamental, geometric definition of $\ln x$ used to build the rest of exponential calculus?
3. How is the general exponential expression $2^\pi$ mathematically defined so it can be rigorously evaluated?
4. What constant factor appears when you take the derivative of $7^x$ that does not appear when you take the derivative of $e^x$?