The Natural Log and Its Properties

You have a number like 50, and you need the exponent that turns e, Euler's number, 2.71828, into 50. That exponent is the natural log of 50, written ln(50). The natural log answers "how many times must I multiply e to get this value?" and it is the inverse of the exponential function e^x. If you are solving a growth or decay problem in chemistry, finance, or computer science, ln is the tool that unwraps the exponential. Treating ln(x) as a variable or swapping it with log(x) as if they were the same function changes your answer by a factor of about 2.3026. They are different functions with different bases, e for natural log, 10 for common log.

By definition, ln(x) = y means e^y = x. So ln(7.389) is about 2 because e² ≈ 7.389. The argument x must be positive, you cannot take the natural log of zero or a negative number in real arithmetic. The domain is x > 0. The result y can be negative: ln(0.5) = -0.6931 because e^{-0.6931} = 0.5. This is the same relationship that defines every logarithm: log_b(x) = y if and only if b^y = x. Natural log is just the version where the base b is e.

Why e Is the Base

e appears naturally in continuous growth processes. If a bank pays 100% annual interest compounded continuously, your money multiplies by e each year. If a radioactive sample decays continuously, the fraction remaining after one half-life is e^{-ln(2)}. The constant e makes the math clean because its rate of change equals its value, the derivative of e^x is e^x, and ln is the inverse that lets you solve for time or rate. No other base gives that property.

On a TI-84 Plus CE calculator, the LOG key returns base 10 and the LN key returns base e. In Microsoft Excel, LOG10(number) gives base 10 and LN(number) gives base e. So far, clear. The trouble starts when you open a textbook. In pure mathematics, "log" often means ln, the natural log. In engineering and chemistry, "log" almost always means log₁₀. The ISO 80000-2:2019 standard tries to fix this by defining lg = log₁₀, ln = logₑ, and lb = log₂. But computer science ignores it: CLRS (Introduction to Algorithms, 4th edition) uses lg to mean log₂. When you see "log" in a formula, check the field. If it is a growth model or calculus text, it is probably ln. If it is a pH or decibel formula, it is base 10. If it is algorithm analysis, it is base 2.

Three values are worth memorising because they anchor every calculation: ln(1) = 0, ln(e) = 1, and ln(e^k) = k. From there, the standard logarithm rules apply unchanged. The product rule: ln(M·N) = ln(M) + ln(N). The quotient rule: ln(M/N) = ln(M) - ln(N). The power rule: ln(M^p) = p · ln(M). The change of base formula lets you compute a natural log in any other base: log_b(x) = ln(x) / ln(b).

Four-figure table values for natural logs are rare, most printed tables give log₁₀. To get ln from a log table, use ln(x) = log₁₀(x) / log₁₀(e). Since log₁₀(e) ≈ 0.4343, you can multiply the common log by 2.3026 to get the natural log. For manual estimation, ln(2) = 0.6931, ln(10) = 2.3026, and ln(0.5) = -0.6931.

Continuous growth models use the form A = P · e^{rt}, where A is the final amount, P is the initial amount, r is the rate, and t is time. To find t, take ln of both sides: ln(A/P) = rt, so t = ln(A/P) / r. For a population that doubles in 10 years, you solve 2 = e^{10r}, so r = ln(2) / 10 ≈ 0.0693 or 6.93% per year.

Radioactive Decay Example

A sample has a half-life of 5 years. The decay constant k satisfies e^{-5k} = 0.5, so k = -ln(0.5) / 5 = 0.1386 per year. To find how long until only 10% remains, solve 0.1 = e^{-0.1386 t}. Taking ln: ln(0.1) = -0.1386 t, so t = -ln(0.1) / 0.1386 ≈ 16.6 years. The failure case here is forgetting the negative sign: ln(0.1) is -2.3026, and dividing by 0.1386 gives -16.6, meaning you would need to reverse time to get 10%. Always check that your t is positive for decay.

Financial Modelling with LN in Excel

In Excel, the LN function takes one argument: LN(number). If you have a growth rate and want the doubling time, enter =LN(2)/r. If you have log₁₀ data and need to convert to natural log, use =LN(number) or =LOG10(number)*LN(10). The LOG function with two arguments, LOG(number, base), works for any base.

The derivative of ln(x) is 1/x for x > 0. This is the reason ln appears so often in calculus, it is the integral of 1/x.The derivative of ln(f(x)) is f'(x) / f(x), by the chain rule.

The derivative of the exponential e^x is e^x, and the derivative of ln(x) is 1/x, which means the two functions are inverses in a way that preserves their own rate of change. If you are solving differential equations in physics or chemistry, these relationships let you separate variables and integrate directly. The power rule for logarithms becomes the logarithm rule for integration: ∫ (1/x) dx = ln|x| + C.