Derivatives · Exponential & Logarithmic

Derivative of log_a(x)

\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

The derivative of a logarithm with base a. Use the change of base formula: log_a(x) = ln(x)/ln(a).

Conditions. x > 0, a > 0, a ≠ 1.

2 worked examples
24 in Derivatives

Worked examples

Find d/dx[log₁₀(x)].

1/(x ln 10)

Answer: 1/(x ln 10) ≈ 0.4343/x

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Find d/dx[log₂(3x)].

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Related formulas

Derivative of ln(x)

See all 24 Derivatives formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Derivative of log_a(x)?

The derivative of a logarithm with base a. Use the change of base formula: log_a(x) = ln(x)/ln(a). It is one of the derivatives formulas in the CalcRef reference.

When does the Derivative of log_a(x) apply?

The Derivative of log_a(x) holds under this condition: x > 0, a > 0, a ≠ 1.

How do you use the Derivative of log_a(x)?

Worked example. Find d/dx[log₁₀(x)]. 1/(x ln 10). Answer: 1/(x ln 10) ≈ 0.4343/x.

Practice

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