# Derivative of log x: Base 10 and Base b, With Examples

> The derivative of log₁₀ x is 1/(x ln 10), and of log_b x is 1/(x ln b). See the change of base proof, chain rule examples and a graph.

Source: https://practicecalculus.com/derivatives/log-x
Updated: 2026-11-06

$$ \frac{d}{dx}\,\log_{10} x = \frac{1}{x\ln 10} $$

## At a glance

- **Derivative of log₁₀ x:** $\dfrac{1}{x\ln 10} \approx \dfrac{0.4343}{x}$
- **General base:** $\dfrac{d}{dx}\log_b x = \dfrac{1}{x\ln b}$
- **Chain rule form:** $\dfrac{d}{dx}\log_b u = \dfrac{u'}{u\ln b}$
- **Change of base:** $\log_b x = \dfrac{\ln x}{\ln b}$
- **Domain:** $x > 0$

## Key takeaways

- The derivative of $\log_{10} x$ is $\frac{1}{x\ln 10}$, and for any base $b$, $\frac{d}{dx}\log_b x = \frac{1}{x\ln b}$.
- The proof is one line: change of base gives $\log_b x = \frac{\ln x}{\ln b}$, a constant times $\ln x$.
- With the chain rule, $\frac{d}{dx}\log_b u = \frac{u'}{u\ln b}$.
- In many calculus books and in most higher math, "log x" means the natural log, so check your course's convention.

The **derivative of $\log_{10} x$ is $\dfrac{1}{x\ln 10}$**. For a log with any base $b$:

$$\frac{d}{dx}\,\log_{10} x = \frac{1}{x\ln 10} \qquad\qquad \frac{d}{dx}\,\log_b x = \frac{1}{x\ln b}$$

It is the [derivative of ln x](/derivatives/ln-x), $\frac{1}{x}$, divided by the constant $\ln b$.

## Which log is "log x"?

The notation depends on where you are:

- On calculators, in chemistry and in most high school courses, $\log x$ means $\log_{10} x$.
- In many calculus books, in higher math and in most programming languages, $\log x$ means $\ln x$.
- In computer science, $\log x$ often means $\log_2 x$.

If you are not sure, look at the answer key: an answer with $\ln 10$ in it means base 10.

## Proof with the change of base formula

The change of base formula turns any log into a natural log:

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

Since $\frac{1}{\ln b}$ is a constant, the [constant multiple rule](/rules/constant-multiple-rule) gives

$$\frac{d}{dx}\log_b x = \frac{1}{\ln b}\cdot\frac{d}{dx}\ln x = \frac{1}{\ln b}\cdot\frac{1}{x} = \frac{1}{x\ln b}$$

For base 10, $\frac{1}{\ln 10} \approx 0.4343$. So a base 10 log grows at about $43\%$ of the rate of the natural log.

## Graph of log₁₀ x and its derivative

```graph
title: 'Graph of y = log₁₀ x and its derivative'
caption: 'The base 10 log grows slowly. Its slope at $x = 1$ is $\frac{1}{\ln 10} \approx 0.434$.'
x: [-0.5, 10]
y: [-2, 2]
functions:
  - expr: ln(x)/ln(10)
    label: y = log₁₀ x
    domain: [0.02, 10]
  - expr: 1/(x*ln(10))
    label: y = 1/(x ln 10) (derivative)
    dashed: true
    domain: [0.25, 10]
points:
  - [1, 0, '(1, 0)']
  - [10, 1, '(10, 1)']
```

## Derivatives for common bases

| Function | Derivative | Approximate |
| --- | --- | --- |
| $\ln x$ | $\frac{1}{x}$ | $\frac{1}{x}$ |
| $\log_{10} x$ | $\frac{1}{x\ln 10}$ | $\frac{0.4343}{x}$ |
| $\log_2 x$ | $\frac{1}{x\ln 2}$ | $\frac{1.4427}{x}$ |
| $\log_3 x$ | $\frac{1}{x\ln 3}$ | $\frac{0.9102}{x}$ |

## Chain rule version and examples

$$\frac{d}{dx}\log_b u = \frac{u'}{u\ln b}$$

**Example 1.** $\frac{d}{dx}\log_{10}(x^2 + 1) = \frac{2x}{(x^2 + 1)\ln 10}$.

**Example 2.** $\frac{d}{dx}\log_2(3x) = \frac{3}{3x\ln 2} = \frac{1}{x\ln 2}$. The $3$ cancels, just like in the [derivative of ln(2x)](/derivatives/ln-2x).

**Example 3: product rule.** For $f(x) = x\log_{10} x$,

$$f'(x) = \log_{10} x + x\cdot\frac{1}{x\ln 10} = \log_{10} x + \frac{1}{\ln 10}$$

**Example 4: decibels.** Sound level is $L = 10\log_{10}\frac{I}{I_0}$. Then $\frac{dL}{dI} = \frac{10}{I\ln 10}$, so each extra unit of intensity matters less as $I$ grows.

## Inverse of aˣ

$\log_b x$ is the inverse of $b^x$, whose derivative is $b^x\ln b$ (see the [derivative of 2ˣ and aˣ](/derivatives/2-to-the-x)). One has $\ln b$ on top and the other has it on the bottom, as the [inverse function rule](/rules/inverse-function-rule) predicts.

## Common mistakes

1. **Writing $\frac{1}{x}$ for base 10.** That is only right for the natural log.
2. **Putting $\log 10$ instead of $\ln 10$.** The factor is always a natural log. ($\log_{10} 10 = 1$, which would wrongly cancel it.)
3. **Writing $\frac{\ln 10}{x}$.** The $\ln b$ goes in the denominator.

For more, see [logarithmic functions](/functions/logarithmic-functions) and [derivatives of exponential and logarithmic functions](/practice-problems/derivatives-of-exponential-and-logarithmic-functions).

## Frequently asked questions

### What is the derivative of log x?

If $\log x$ means base 10, the derivative is $\frac{1}{x\ln 10}$. If it means the natural log, as in many advanced texts, the derivative is $\frac{1}{x}$.

### What is the derivative of log base 2 of x?

It is $\frac{1}{x\ln 2}$, which is about $\frac{1.443}{x}$.

### Why is there an ln in the derivative of a base 10 log?

Because $\log_{10} x = \frac{\ln x}{\ln 10}$. The derivative of $\ln x$ is $\frac{1}{x}$, and the constant $\frac{1}{\ln 10}$ just comes along.

### What is the derivative of log(x² + 1)?

With base 10, it is $\frac{2x}{(x^2 + 1)\ln 10}$.