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Exponential and log derivatives

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.

Exponential and log derivatives Updated 2 min read

At a glance

\frac{d}{dx}\,\log_{10} x = \frac{1}{x\ln 10}
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
On this page
  1. Which log is "log x"?
  2. Proof with the change of base formula
  3. Graph of log₁₀ x and its derivative
  4. Derivatives for common bases
  5. Chain rule version and examples
  6. Inverse of aˣ
  7. Common mistakes

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, \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 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 of y = log₁₀ x and its derivative
The base 10 log grows slowly. Its slope at x = 1 is \frac{1}{\ln 10} \approx 0.434.

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).

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). One has \ln b on top and the other has it on the bottom, as the 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 and derivatives of exponential and logarithmic functions.

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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}.

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Related derivatives

Derivative of ln x \frac{d}{dx}\,\ln x = \frac{1}{x} The derivative of ln x is 1/x. See why, with a proof from eʸ = x, the chain rule version for ln(u), a graph and worked examples like ln(2x) and ln(x²+1). Derivative of 2^x \frac{d}{dx}\,2^x = 2^x\ln 2 The derivative of 2ˣ is 2ˣ ln 2, and of aˣ is aˣ ln a. See the proof by rewriting aˣ as e^(x ln a), chain rule examples and a graph. Derivative of ln(2x) \frac{d}{dx}\,\ln(2x) = \frac{1}{x} The derivative of ln(2x) is 1/x, the same as ln x. See two proofs, a graph of why the slopes match, and how ln(2x + 1) is different.