Hard

The Derivative of General Logarithmic Functions

Simple Explanation

For a base b other than e, the derivative of log_b(x) picks up an extra constant factor in the denominator: d/dx[log_b x] = 1 / (x · ln b).

Why Do We Need It?

Just as with general exponential functions, most real logarithmic scales (like the base-10 Richter or pH scales) do not use base e directly — this formula handles any base correctly.

Formula

The Derivative of a General Logarithmic Function

d/dx[log_b x] = 1 / (x · ln b)

For any base b, the derivative of log_b(x) includes an extra factor of 1/ln(b) — this reduces to the simple 1/x case exactly when b=e, since ln(e)=1.

b: — the base of the logarithm

When to use it: Whenever differentiating a logarithmic function with a base other than e.

Worked Example

Differentiate a general logarithmic function

Find the derivative of f(x)=log₂(x), and evaluate f'(1). (ln 2 ≈ 0.6931)

Why Does This Work?

Using the change-of-base identity, log_b(x) = ln(x)/ln(b) — since ln(b) is just a constant, differentiating gives (1/ln b)·d/dx[ln x] = (1/ln b)·(1/x) = 1/(x·ln b), exactly the stated formula.

Real-Life Example

Finding the rate of change on a base-10 measurement scale

The pH scale (base-10 logarithmic) measures acidity, and chemists sometimes need to know how quickly pH changes as hydrogen ion concentration changes.

The general logarithm derivative rule, using base 10, gives exactly this rate of change.

Practice

For f(x)=log_b(x) with f'(x)=1/(x·ln b), if b=e, what does ln(b) equal (so that the formula simplifies to 1/x)?

Hard

Common mistake

Forgetting the ln(b) factor in the denominator for bases other than e — only natural log (base e) gives the simple 1/x result.

Quick Review

d/dx[log_b x] = 1 / (x · ln b).
Reduces to the simple 1/x case exactly when b=e.
Derived using the change-of-base identity, log_b(x) = ln(x)/ln(b).