The Natural Logarithm

6 min read

Why Base $e$?

Of all the possible bases for logarithms (2, 10, 7.3, ...), one stands out: the number $e \approx 2.71828$. The logarithm with base $e$ is so important that it gets its own name and notation:

$$\ln x = \log_e x$$

But why is base $e$ special? The full answer will become clear when you study derivatives in Section 6.4, but here’s a preview: the derivative of $\ln x$ is simply $\frac{1}{x}$, with no messy constants. This makes $\ln x$ the “natural” choice for calculus.

For now: Think of $\ln$ as “the logarithm that calculus likes best.”

Prerequisite Map

Before You Start

Self-check: Can you answer these questions? If not, review the linked prerequisites first.

Quick Reference

Key Concepts

Definition of the Natural Logarithm

The natural logarithm is the logarithm with base $e$:

$$\boxed{\ln x = \log_e x}$$

By the definition of logarithm:

$$\boxed{\ln x = y \quad \Longleftrightarrow \quad e^y = x}$$

In words: $\ln x$ is the power to which $e$ must be raised to get $x$.

Cancellation Equations for $\ln$ and $e^x$

Since $\ln x$ and $e^x$ are inverses:

$$\boxed{\ln(e^x) = x \quad \text{for all real } x}$$

$$\boxed{e^{\ln x} = x \quad \text{for all } x > 0}$$

These are your main tools for solving equations involving $e$ and $\ln$.

Special Values

Domain, Range, and Graph

    y                              y
    |     y = e^x                  |     y = ln(x)
    |        /                     |            ___
  3 +       /                    2 +        ___/
    |      /                       |    ___/
  1 +----+/                      0 +---+----------→ x
    |   /                          |  /|
    |  /                           | / | (1, 0)
    | /                          -2+/  |
    +/------------→ x              |   |
        1   2                      |   x = 0 (asymptote)

Key observations:

• The graph of $y = \ln x$ is the reflection of $y = e^x$ about the line $y = x$
• $\ln x$ crosses the $x$-axis at $x = 1$ (since $\ln 1 = 0$)
• The $y$-axis ($x = 0$) is a vertical asymptote
• $\ln x$ grows without bound as $x \to \infty$, but very slowly

Limits of the Natural Logarithm

$$\lim_{x \to \infty} \ln x = \infty \qquad \text{(but grows very slowly)}$$

$$\lim_{x \to 0^+} \ln x = -\infty \qquad \text{(vertical asymptote)}$$

How slowly does $\ln x$ grow?

To get $\ln x = 20$, you need $x = e^{20} \approx 485{,}000{,}000$. That’s nearly half a billion!

The Power Rule: $x^r = e^{r \ln x}$

This identity connects powers to exponentials:

$$\boxed{x^r = e^{r \ln x} \quad \text{for } x > 0}$$

Why it works: Start with the second cancellation equation $e^{\ln x} = x$. Raise both sides to the power $r$:

$$(e^{\ln x})^r = x^r$$ $$e^{r \ln x} = x^r$$

This identity is crucial for logarithmic differentiation (Section 6.4).

The Change of Base Formula

Any logarithm can be expressed in terms of natural logarithms:

$$\boxed{\log_b x = \frac{\ln x}{\ln b}}$$

Why it works: Let $y = \log_b x$. Then $b^y = x$. Take natural log of both sides:

$$\ln(b^y) = \ln x$$ $$y \ln b = \ln x$$ $$y = \frac{\ln x}{\ln b}$$

Practical use: Your calculator has an $\ln$ button but probably not a $\log_7$ button. To compute $\log_7 50$:

$$\log_7 50 = \frac{\ln 50}{\ln 7} \approx \frac{3.912}{1.946} \approx 2.01$$

Solving Equations with $\ln$ and $e^x$

Strategy: Use the inverse relationship. Apply $e^{(\cdot)}$ to “undo” $\ln$, or apply $\ln$ to “undo” $e^{(\cdot)}$.

Example 1: Solve $\ln x = 5$.

Apply $e^{(\cdot)}$ to both sides: $$e^{\ln x} = e^5$$ $$x = e^5$$

Example 2: Solve $e^{5-3x} = 10$.

Take natural log of both sides: $$\ln(e^{5-3x}) = \ln 10$$ $$5 - 3x = \ln 10$$ $$x = \frac{5 - \ln 10}{3}$$

Common Pitfalls

Practice Problems

Common Misconceptions

Mastery Checklist

• I can state the definition: $\ln x = y$ means $e^y = x$
• I can evaluate $\ln e^k$ and $e^{\ln k}$ using cancellation equations
• I know the special values: $\ln 1 = 0$, $\ln e = 1$
• I can sketch the graph of $y = \ln x$ showing key features
• I can find limits involving $\ln x$ as $x \to 0^+$ and $x \to \infty$
• I can solve equations like $\ln x = k$ and $e^{f(x)} = k$
• I can use the change of base formula: $\log_b x = \frac{\ln x}{\ln b}$
• I can apply graph transformations to $y = \ln x$

Mental Model

$e^x$ and $\ln x$ are Opposites:

Think of $e^x$ as a “growth machine” and $\ln x$ as an “un-growth machine.”

• $e^x$ asks: “If I grow at rate $x$, what do I become?”
• $\ln x$ asks: “What growth rate produced $x$?”

When you compose them, they cancel:

• Feed $x$ into the growth machine, then un-grow: you get $x$ back
• Un-grow $x$ first, then grow at that rate: you get $x$ back

This is the essence of inverse functions.

Connections

Looking back:

• Logarithm Definition: $\ln$ is just $\log_e$; everything from the general definition applies
• Exponential functions: $e^x$ is the inverse; their graphs are reflections

Looking ahead:

• Laws of Logarithms: How $\ln$ interacts with products, quotients, and powers
• Derivatives (Section 6.4): The derivative of $\ln x$ is $\frac{1}{x}$, remarkably simple
• Integration (Section 6.4): $\int \frac{1}{x}\,dx = \ln\vert x\vert + C$

Why “natural”?

The natural logarithm appears everywhere in calculus:

• The derivative of $\ln x$ is $\frac{1}{x}$ (no extra constants)
• The integral of $\frac{1}{x}$ is $\ln\vert x\vert $
• Compound interest formula involves $e$ and $\ln$
• Solutions to differential equations naturally involve $e^x$ and $\ln x$

No other base produces formulas this clean.

Last updated: 2026-01-23