# Natural Logarithm ln(x): Graph, Rules, Derivative and Integral

> The natural logarithm ln x: its graph, log rules, why its derivative is 1/x, the integral x ln x - x + C, and worked calculus examples.

Source: https://practicecalculus.com/functions/natural-logarithm
Updated: 2026-11-03

$$ f(x) = \ln x $$

## At a glance

- **Domain:** $(0, \infty)$
- **Range:** $(-\infty, \infty)$
- **Derivative:** $\frac{1}{x}$
- **Integral:** $x\ln x - x + C$
- **Key values:** $\ln 1 = 0$, $\ln e = 1$
- **Inverse:** $e^x$
- **Asymptote:** $x = 0$

## Key takeaways

- The natural logarithm $\ln x$ is the inverse of $e^x$: $\ln x = y$ means $e^y = x$.
- Its derivative is $\frac{1}{x}$, and with the chain rule $\frac{d}{dx}\ln g(x) = \frac{g'(x)}{g(x)}$.
- Its integral, found by integration by parts, is $x\ln x - x + C$.
- Log rules turn products into sums and powers into multiples, which simplifies hard derivatives.

The **natural logarithm** $\ln x$ answers the question "what power of $e$ gives $x$?". It is the inverse of the [natural exponential function](/functions/natural-exponential-function), and its derivative is the remarkably simple $\frac{1}{x}$, which makes it the logarithm of choice in calculus.

## Definition and formula

$$\ln x = y \iff e^y = x, \qquad x > 0$$

Equivalently, $\ln x$ is the area under the [reciprocal function](/functions/reciprocal-function) from $1$ to $x$:

$$\ln x = \int_1^x \frac{1}{t}\,dt$$

## Graph of ln x

```graph
title: 'Graph of y = ln(x) and its inverse y = eˣ'
caption: 'The natural log is the reflection of $e^x$ across the line $y = x$. It crosses the $x$-axis at $(1, 0)$ and has a vertical asymptote at $x = 0$.'
x: [-3, 6]
y: [-3, 6]
functions:
  - expr: ln(x)
    label: y = ln x
  - expr: e^x
    label: y = eˣ
  - expr: x
    label: y = x
    dashed: true
points:
  - [1, 0, '(1, 0)']
  - [2.718, 1, '(e, 1)']
```

## Key properties

- **Domain** $(0, \infty)$, **range** all reals.
- **Key values:** $\ln 1 = 0$, $\ln e = 1$, $\ln e^k = k$.
- **Increasing** everywhere ($\frac{1}{x} > 0$) and **concave down** ($-\frac{1}{x^2} < 0$).
- **Vertical asymptote** $x = 0$: $\lim_{x \to 0^+}\ln x = -\infty$. Also $\lim_{x \to \infty}\ln x = \infty$, but very slowly; $\ln x$ grows slower than any positive power of $x$.
- **Inverse identities:** $e^{\ln x} = x$ for $x > 0$ and $\ln e^x = x$ for all $x$.

## Log rules

| Rule | Formula |
| --- | --- |
| Product | $\ln(ab) = \ln a + \ln b$ |
| Quotient | $\ln\frac{a}{b} = \ln a - \ln b$ |
| Power | $\ln(a^k) = k\ln a$ |
| Change of base | $\log_b x = \frac{\ln x}{\ln b}$ |

These hold for $a, b > 0$. There is no rule for $\ln(a + b)$.

## Derivative of ln x

$$\frac{d}{dx}\ln x = \frac{1}{x}$$

Proof sketch: if $y = \ln x$ then $e^y = x$. Differentiating implicitly gives $e^y y' = 1$, so $y' = \frac{1}{e^y} = \frac{1}{x}$. With the [chain rule](/rules/chain-rule):

$$\frac{d}{dx}\ln g(x) = \frac{g'(x)}{g(x)}$$

## Integral of ln x

Using [integration by parts](/blog/integration-by-parts) with $u = \ln x$, $dv = dx$, so $du = \frac{1}{x}\,dx$ and $v = x$:

$$\int \ln x\,dx = x\ln x - \int x \cdot \frac{1}{x}\,dx = x\ln x - x + C$$

The reverse direction is the more common one: $\int \frac{1}{x}\,dx = \ln|x| + C$.

## Worked examples

**Example 1: chain rule.** $\frac{d}{dx}\ln(x^2 + 1) = \frac{2x}{x^2 + 1}$.

**Example 2: simplify first.** $\frac{d}{dx}\ln\frac{x^3}{x + 1} = \frac{d}{dx}\left[3\ln x - \ln(x + 1)\right] = \frac{3}{x} - \frac{1}{x + 1}$.

**Example 3: product rule.** $\frac{d}{dx}(x^2\ln x) = 2x\ln x + x$.

**Example 4: definite integral.** $\int_1^e \ln x\,dx = \left[x\ln x - x\right]_1^e = (e - e) - (0 - 1) = 1$.

**Example 5: logarithmic integral.** With $u = x^2 + 4$, $\int \frac{2x}{x^2 + 4}\,dx = \ln(x^2 + 4) + C$. No absolute value is needed since $x^2 + 4 > 0$.

## Common mistakes

1. **Splitting a log of a sum.** $\ln(x + 1) \ne \ln x + \ln 1$.
2. **Writing $\frac{d}{dx}\ln(5x) = \frac{1}{5x}$.** The chain rule gives $\frac{5}{5x} = \frac{1}{x}$.
3. **Writing $\int \ln x\,dx = \frac{1}{x} + C$.** That is the derivative, not the integral.
4. **Confusing $\ln x^2$ with $(\ln x)^2$.** The first is $2\ln x$; the second is a square.
5. **Taking $\ln$ of $0$ or a negative number.** Check the domain, especially after a substitution.

## How it connects

Taking $\ln$ of both sides turns hard products and powers into sums, the idea behind [logarithmic differentiation](/rules/logarithmic-differentiation). Logs with other bases are just rescaled natural logs; see [logarithmic functions](/functions/logarithmic-functions). For more worked practice, read the guide to [derivatives of exponential and logarithmic functions](/blog/derivatives-of-exponential-and-logarithmic-functions).

## Frequently asked questions

### What is the derivative of ln x?

The derivative of $\ln x$ is $\frac{1}{x}$ for $x > 0$. More generally, $\frac{d}{dx}\ln|x| = \frac{1}{x}$ for every $x \ne 0$.

### What is the integral of ln x?

Use integration by parts with $u = \ln x$ and $dv = dx$: $\int \ln x\,dx = x\ln x - x + C$.

### What is the difference between ln and log?

$\ln x$ is the logarithm with base $e \approx 2.718$. In most calculus books $\log x$ means base $10$, though some advanced texts use $\log$ for base $e$ as well.

### What is the derivative of ln(2x)?

By the chain rule, $\frac{d}{dx}\ln(2x) = \frac{2}{2x} = \frac{1}{x}$. Equivalently, $\ln(2x) = \ln 2 + \ln x$, and the constant $\ln 2$ has derivative $0$.

### Why is ln of a negative number undefined?

Because $e^y$ is positive for every real $y$, no real power of $e$ equals a negative number or zero. So $\ln x$ is only defined for $x > 0$.