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Derivative of Logarithm (base a)

Complete guide with formula, proof, examples, and graph.

Quick Answer

ddxlog⁡a(x)=1xln⁡(a)\frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}

The derivative of \log_a(x) is:

Proof / Derivation

Step-by-step derivation of the derivative formula.

Use the change-of-base formula to express the logarithm in terms of natural log.

log⁡a(x)=ln⁡(x)ln⁡(a)\log_a(x) = \frac{\ln(x)}{\ln(a)}

Note that ln(a) is a constant factor and can be pulled out.

Sinceln⁡(a)isconstant:Since \ln(a) is constant:

Apply the constant multiple rule.

ddxlog⁡a(x)=1ln⁡(a)⋅ddxln⁡(x)\frac{d}{dx}\log_a(x) = \frac{1}{\ln(a)} \cdot \frac{d}{dx}\ln(x)

Substitute d/dx[ln(x)] = 1/x to obtain the final result.

=1ln⁡(a)⋅1x=1xln⁡(a)= \frac{1}{\ln(a)} \cdot \frac{1}{x} = \frac{1}{x\ln(a)}

Graph

Visualization of Logarithm (base a) and its derivative.

f(x) = \log_a(x)

f(x)=log⁡a(x)f(x) = \log_a(x)

f'(x) = \frac{1}{x\ln(a)}

f′(x)=1xln⁡(a)f'(x) = \frac{1}{x\ln(a)}
Domain: (0,+∞),a>0,a≠1(0, +\infty), a > 0, a \neq 1Range: (−∞,+∞)(-\infty, +\infty)

Worked Examples

Step-by-step solutions using the chain rule and other techniques.

Find: ddxlog⁡10(x)\frac{d}{dx}\log_{10}(x)

Solution: 1xln⁡(10)\frac{1}{x\ln(10)}

1.a=10,applyformulaa = 10, apply formula
2.=1xln⁡(10)= \frac{1}{x\ln(10)}

Find: ddxlog⁡2(x3)\frac{d}{dx}\log_2(x^3)

Solution: 3xln⁡(2)\frac{3}{x\ln(2)}

1.Chainrule:u=x3,u′=3x2Chain rule: u = x^3, u' = 3x^2
2.=1x3ln⁡(2)⋅3x2=3xln⁡(2)= \frac{1}{x^3\ln(2)} \cdot 3x^2 = \frac{3}{x\ln(2)}

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Frequently Asked Questions

What is the derivative of Logarithm (base a)?

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The derivative of log⁡a(x)\log_a(x) is 1xln⁡(a)\frac{1}{x\ln(a)}. This is one of the fundamental derivatives in calculus that you should memorize.

How do you prove the derivative of Logarithm (base a)?

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The proof uses the limit definition of the derivative. See the Proof section above for the complete step-by-step derivation.

Is the derivative of Logarithm (base a) always the same?

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Yes, the derivative formula for log⁡a(x)\log_a(x) is constant — it does not depend on x. However, when composed with inner functions (e.g., log⁡a(x)\log_a(x) of u(x)), the chain rule applies.

Where is the derivative of Logarithm (base a) undefined?

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The derivative is undefined where the original function is not differentiable. Check the domain section for details.

Why is the derivative of Logarithm (base a) important?

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This derivative appears frequently in physics (wave motion), engineering (signal processing), economics (oscillating models), and many other fields involving periodic or growth phenomena.