Logarithmic Differentiation

Logarithmic differentiation is a powerful technique that simplifies the process of differentiating complex functions — particularly those involving products, quotients, or variable exponents — by first taking the natural logarithm of both sides of an equation. Because the logarithm converts multiplication into addition and exponentiation into multiplication, it dramatically reduces algebraic complexity before any derivative is computed. The method is especially indispensable when dealing with functions of the form y = f(x)^{g(x)}, where both the base and the exponent are functions of x. Standard rules like the power rule or exponential rule alone cannot handle such expressions, but logarithmic differentiation handles them elegantly in a few systematic steps. It also shines when differentiating long products or quotients that would otherwise require repeated applications of the product and quotient rules. Mastering logarithmic differentiation deepens your understanding of implicit differentiation, the chain rule, and logarithmic identities — all of which are core tools in calculus. Beyond the classroom, this technique appears in physics, economics, and engineering wherever growth rates, elasticities, and power-law relationships arise.

Key takeaways
  • Take the Natural Logarithm of Both Sides
  • Expand Using Logarithm Laws
  • Differentiate Both Sides Implicitly with Respect to x
  • Solve for dy/dx
  • Special Case — Variable Exponents

Take the Natural Logarithm of Both Sides

Begin with y = f(x). Apply ln to both sides: ln(y) = ln(f(x)). This is valid whenever f(x) > 0. If f(x) can be negative, use ln|y| = ln|f(x)| and the result is the same after differentiation.

Expand Using Logarithm Laws

Use the three key log laws to simplify the right side: ln(AB) = ln A + ln B, ln(A/B) = ln A - ln B, and ln(A^k) = k ln A. For example, if y = x^3 · e^{2x} / sin(x), then ln(y) = 3 ln(x) + 2x - ln(sin(x)).

Differentiate Both Sides Implicitly with Respect to x

Differentiate ln(y) with respect to x using the chain rule: d/dx[ln(y)] = (1/y)(dy/dx). The right side is differentiated term by term, which is now straightforward since products have become sums.

Solve for dy/dx

Multiply both sides by y to isolate dy/dx: dy/dx = y · [d/dx of the simplified right side]. Finally, substitute back the original expression for y in terms of x so that the answer is entirely in terms of x.

Special Case — Variable Exponents

For y = [f(x)]^{g(x)}, taking ln gives ln(y) = g(x) · ln(f(x)). Differentiating: (1/y)(dy/dx) = g'(x) ln(f(x)) + g(x) · f'(x)/f(x). Multiplying by y gives: dy/dx = [f(x)]^{g(x)} · [g'(x) ln(f(x)) + g(x) f'(x)/f(x)]. This formula covers cases impossible with simpler rules.

The Key Equation

dydx=y⋅ddx[ln⁡y]=f(x)⋅ddx[ln⁡f(x)]\frac{dy}{dx} = y \cdot \frac{d}{dx}\left[\ln y\right] = f(x)\cdot\frac{d}{dx}\left[\ln f(x)\right]
  • y — Function (dimensionless): The original function y = f(x) whose derivative we seek.
  • \ln y — Natural logarithm of y (dimensionless): The natural log (base e) applied to y; its derivative with respect to x is (1/y)(dy/dx) by the chain rule.
  • \frac{dy}{dx} — Derivative of y (depends on context): The rate of change of y with respect to x — the quantity we are solving for.
  • \frac{d}{dx}[\ln f(x)] — Logarithmic derivative (per unit of x): The derivative of ln f(x), equal to f'(x)/f(x); this is called the logarithmic derivative of f.

Worked Example

Problem. Find dy/dx if y = x^x for x > 0.

dydx=xx(ln⁡x+1)\frac{dy}{dx} = x^x\left(\ln x + 1\right)

Answer. dy/dx = x^x (ln x + 1)

Figure
Diagram
Logarithmic Differentiation: Step-by-Step Flowy = f(x)ln(y) = ln f(x) [apply log laws]⅓/y · dy/dx = d/dx [simplified RHS]dy/dx = y · [derivative of RHS]Substitute y = f(x) to get final answer
Origins · 1 minJohann Bernoulli and how this was discovered
JB
Johann Bernoulli
1667–1748 · Swiss

Johann Bernoulli was one of the early masters of differential calculus following Newton and Leibniz. He was among the first mathematicians to systematically apply logarithmic techniques to differentiation, recognising that the natural logarithm could convert multiplicative structures — hard to differentiate directly — into additive ones amenable to term-by-term differentiation. His correspondence with Leibniz helped standardise notation and methods that underpin logarithmic differentiation as taught today. Bernoulli was famously competitive with his own brother Jakob and even with his son Daniel, publicly disputing priority for mathematical discoveries — a reminder that brilliant mathematicians are very human!

  • The Chain Rule — Logarithmic differentiation relies on the chain rule at every step — specifically to differentiate ln(y) as (1/y)(dy/dx) when y is itself a function of x.
  • Implicit Differentiation — After taking ln of both sides, the technique proceeds exactly like implicit differentiation, treating y as a function of x and differentiating both sides with respect to x.
  • The Product Rule — Logarithmic differentiation is often used as an alternative to repeated application of the product rule, converting products into sums via ln(AB) = ln A + ln B.
  • Laws of Logarithms — The three logarithm laws — product, quotient, and power — are essential prerequisites; without them, the algebraic simplification step of logarithmic differentiation cannot be performed.