Natural Logarithm and the ln Function

The natural logarithm ln x is the logarithm with base e ≈ 2.71828, and it is the inverse of the exponential function e^x: ln(e^x) = x and e^{ln x} = x. In brief: ln 1 = 0, ln e = 1, ln 10 ≈ 2.3026 and ln 2 ≈ 0.6931. The derivative of ln x is 1/x, and the function appears in continuous growth: a quantity growing at the continuous rate r takes t = ln 2/r units of time to double.

The natural logarithm is the preferred logarithm in mathematics and science, because it has the simplest derivative and ties together growth, area under the hyperbola and complex numbers. This guide explains the definition, the key values and properties, how ln solves exponential equations, the derivative and integral, the relation to the area under y = 1/x, and gives a table of values with worked applications.

What is the natural logarithm?

ln x = log_e x, where e is the base of natural exponential growth. So y = ln x means e^y = x. The domain is x > 0 and the range is all real numbers. The graph passes through (1, 0) and (e, 1), has the y-axis as the vertical asymptote and increases without bound, but slowly.

Another definition from calculus: ln x is the area under the curve y = 1/t between t = 1 and t = x, that is ln x = ∫₁^x (1/t) dt. For x > 1 this area is positive, and for 0 < x < 1 it counts as negative. The number e is then the value of x for which the area equals 1.

What are its properties?

All the usual properties of logarithms hold:

• ln(xy) = ln x + ln y, ln(x/y) = ln x − ln y, ln(xⁿ) = n ln x.
• ln 1 = 0, ln e = 1, ln(1/x) = −ln x.
• e^{ln x} = x for x > 0 and ln(e^x) = x for all x.
• ln x < 0 for 0 < x < 1 and ln x > 0 for x > 1.
• ln x ≤ x − 1 for all x > 0, with equality only at x = 1. For small h, ln(1 + h) ≈ h.
• The change of base: log_b x = ln x/ln b.

Check of the inequality: ln 2 = 0.693 ≤ 1 ✓ and ln 0.5 = −0.693 ≤ −0.5 ✓. And the approximation: ln 1.05 = 0.04879 ≈ 0.05.

How do you use ln to solve equations?

Apply ln to both sides of an exponential equation. The power rule brings down the exponent.

Example 1: e^2x = 7

Take ln: 2x = ln 7, so x = (ln 7)/2 ≈ 1.9459/2 ≈ 0.973.

Example 2: 3 · 5^x = 60

5^x = 20, so x ln 5 = ln 20 and x = ln 20/ln 5 = 2.9957/1.6094 ≈ 1.861.

Example 3: ln(x − 1) + ln(x + 2) = ln 4

Domain: x > 1. Combine: (x − 1)(x + 2) = 4, so x² + x − 6 = 0, giving (x + 3)(x − 2) = 0. Only x = 2 is in the domain. Check: ln 1 + ln 4 = ln 4 ✓.

Example 4: e^x − 6 + 8e^−x = 0

Multiply by e^x: e^2x − 6e^x + 8 = 0. Let u = e^x: (u − 2)(u − 4) = 0, so u = 2 or 4 and x = ln 2 or ln 4 = 2 ln 2.

What are the derivative and the integral of ln x?

The derivative of ln x is 1/x for x > 0. This is why the natural logarithm is natural: the slope at x = 1 is 1. Using the chain rule, (ln f(x))′ = f′(x)/f(x). Examples: (ln(3x))′ = 1/x, (ln(x² + 1))′ = 2x/(x² + 1).

The integral ∫ (1/x) dx = ln|x| + C fills the gap in the power rule for integration: the antiderivative of xⁿ is xⁿ⁺¹/(n + 1), which fails for n = −1. The antiderivative of ln x is x ln x − x + C, found by integration by parts.

Logarithmic differentiation uses ln to differentiate products and powers: for y = x^x, ln y = x ln x, so y′/y = ln x + 1 and y′ = x^x(ln x + 1).

Table of values of ln x

Worked applications

Example 5: doubling time

A quantity grows continuously at 6% per unit of time: N = N₀e^0.06t. It doubles when e^0.06t = 2, so t = ln 2/0.06 = 11.55. The rule of 70 gives 70/6 ≈ 11.7, a good approximation.

Example 6: radioactive decay

N(t) = N₀e^−kt with a half-life of 5,730 years (carbon-14): k = ln 2/5730 ≈ 1.21 × 10⁻⁴ per year. A sample with 25% of the original carbon-14 has age t = ln 4/k = 2 · 5730 = 11,460 years.

Example 7: continuous compounding

How long does 2,000 dollars at 3.5% continuous interest need to reach 5,000? 2000e^0.035t = 5000, so e^0.035t = 2.5 and t = ln 2.5/0.035 = 0.9163/0.035 ≈ 26.2 years.

Example 8: time of cooling

A cup of coffee cools following T(t) = 20 + 60e^−0.05t (°C, t in minutes). It reaches 40 °C when 60e^−0.05t = 20, so e^−0.05t = 1/3 and t = ln 3/0.05 = 21.97 minutes.

Where does the natural logarithm come from?

The natural logarithm grew out of the work of Napier, who defined logarithms by a kinematic argument. Its connection with the area under the hyperbola was found by Gregoire de Saint-Vincent and Alphonse de Sarasa in the 1640s, and Nicholas Mercator published the series ln(1 + x) = x − x²/2 + x³/3 − … in 1668. Euler identified e as the base and fixed the modern notation. The abbreviation ln is attributed to Irving Stringham, who used it in 1893.

Where is it used today?

The natural logarithm solves problems of continuous growth and decay, appears in the entropy of physical systems and information (measured in nats), and in the normal distribution of statistics. It is used in logistic regression, in the Black–Scholes model in finance and in the definition of the Lyapunov exponent in chaos theory. Taking the logarithm of data with multiplicative changes turns them into additive changes.

Which mistakes are common?

Confusing ln with log

ln is base e and log is usually base 10 on calculators (but in higher mathematics log often means ln). Check the convention.

Thinking ln(x + y) = ln x + ln y

Only products can be split: ln(xy) = ln x + ln y.

Forgetting the absolute value in the antiderivative

∫ 1/x dx = ln|x| + C, since 1/x is defined for negative x as well.

Taking ln of a negative number

ln x is defined only for x > 0. For ln(x − 3), the domain is x > 3.