CalcRef

Derivatives · Exponential & Logarithmic

Derivative of log_a(x)

ddx[log⁡ax]=1xln⁡a\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.

Worked examples

Find d/dx[log₁₀(x)].
  1. 1/(x ln 10)

Answer: 1/(x ln 10) ≈ 0.4343/x

Find d/dx[log₂(3x)].
  1. Chain rule: 1/((3x) ln 2) · 3 = 3/(3x ln 2) = 1/(x ln 2)

Answer: 1/(x ln 2)

Related formulas

Derivative of ln(x)

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