The Natural Logarithm

The natural logarithm ln(x) is the inverse of eˣ. It is uniquely important in calculus because it solves the one integration problem the power rule cannot handle: ∫ 1/x dx.

∫ 1/x dx = ln|x| + C

Derivatives of Logarithms

d/dx [ln(x)] = 1/x d/dx [ln(f(x))] = f'(x)/f(x) (chain rule) d/dx [log_a(x)] = 1/(x·ln(a))
📋 Worked Examples
1d/dx[ln(x²+1)] = 2x/(x²+1)
2d/dx[ln(sin x)] = cos(x)/sin(x) = cot(x)
3∫ tan(x) dx = ∫ sin/cos dx = −ln|cos x| + C

Logarithmic Differentiation

For complicated products, quotients, or functions with variable exponents, take ln of both sides first, then differentiate implicitly:

📋 Find d/dx[xˣ]
Step 1Let y = xˣ. Take ln: ln y = x·ln x
Step 2Differentiate: y'/y = ln x + 1
Step 3y' = xˣ(ln x + 1)

Frequently Asked Questions

Why ln|x| and not just ln(x)?▾
Because ln(x) is only defined for x > 0, but 1/x is defined for all x ≠ 0. The absolute value |x| extends the antiderivative to negative x as well. For definite integrals where x stays positive, you can drop the absolute value.
When do I use logarithmic differentiation?▾
When the function is a product/quotient of many terms (logarithms turn multiplication into addition, simplifying the derivative), or when the variable appears in both base and exponent, like xˣ or x^(sin x).

Why the Natural Log?

You might wonder why we prefer ln(x) over log₁₀(x) in calculus. The reason is purely the derivative: d/dx[ln(x)] = 1/x, a clean result with no extra constants. For log₁₀(x), the derivative is 1/(x·ln 10) — the extra ln 10 factor appears in every calculation and makes everything messier. The natural logarithm is "natural" precisely because it produces the cleanest calculus.

The Missing Power Rule Case

The power rule says ∫xⁿ dx = xⁿ⁺¹/(n+1) + C — but this fails for n = −1 (division by zero). The natural log fills exactly this gap:

∫ 1/x dx = ln|x| + C ← the case the power rule cannot handle

This is why ln(x) is so central to integration. Any time a rational function is integrated, the logarithm appears in the answer — through partial fractions, through substitution, or directly.

More Derivative Examples

📋 Worked examples
d/dx[ln(3x)]= 1/(3x) · 3 = 1/x (the 3 always cancels for linear arguments)
d/dx[ln(x³+1)]= 3x²/(x³+1)
d/dx[ln(sin x)]= cos(x)/sin(x) = cot(x)
d/dx[x·ln(x)]= ln(x) + x·(1/x) = ln(x) + 1 (product rule)
d/dx[ln(eˣ)]= x → d/dx = 1 (ln and exp cancel)

Integration with Logarithms

📋 Integrals involving ln
∫ tan(x) dx= ∫ sin/cos dx. Let u = cos x → −ln|cos x| + C = ln|sec x| + C
∫ 1/(2x+1) dx= ½ ln|2x+1| + C (u = 2x+1)
∫₁ᵉ ln(x) dxIBP: u=ln x, dv=dx → [x·ln x − x]₁ᵉ = (e−e)−(0−1) = 1

Log Laws You Need for Calculus

ln(ab) = ln(a) + ln(b) ln(a/b) = ln(a) − ln(b) ln(aⁿ) = n·ln(a) ln(eˣ) = x and e^(ln x) = x ← inverse functions

These laws are used constantly in calculus — to simplify before differentiating, to apply logarithmic differentiation, and to solve equations involving exponentials.

Logarithmic Differentiation — When to Use It

Take ln of both sides before differentiating when: (1) the function has many factors (log turns products into sums), or (2) the variable appears in both the base and exponent.

📋 Differentiating xˢⁱⁿˣ
Take lnln y = sin(x)·ln(x)
Differentiatey'/y = cos(x)·ln(x) + sin(x)/x
Multiply by yy' = xˢⁱⁿˣ [cos(x)·ln(x) + sin(x)/x]
AM
Dr. Aisha Malik, PhD Mathematics
Senior Lecturer in Applied Mathematics · 12 years teaching calculus
Dr. Malik holds a PhD in Applied Mathematics from the University of Edinburgh and has taught calculus to over 4,000 students. She has reviewed this article for mathematical accuracy and pedagogical clarity.
Technically reviewed by: Prof. James Chen, Stanford Mathematics Department · April 2026