Natural Logarithm and e

Core Idea

The number e (approximately 2.71828) is the base of the natural exponential function and natural logarithm. It arises naturally from continuous compounding: as n approaches infinity, (1 + 1/n)ⁿ approaches e. The natural logarithm ln(x) = log_e(x) is the inverse of e^x. The pair (e^x, ln(x)) is fundamental in calculus because the derivative of e^x is e^x itself. Continuous growth/decay is modeled by A = A_0 * e^kt.

How It's Best Learned

Motivate e through compound interest: show that compounding more frequently approaches a limit. Define e as that limit. Practice converting between e^x = y and ln(y) = x. Solve equations involving e^x and ln(x). Compare natural log with common log. Emphasize that e is just a number (albeit irrational and transcendental), not a variable.

Common Misconceptions

• Thinking e is a variable rather than a specific constant.
• Confusing ln(x) with log(x) (ln uses base e, log typically uses base 10).
• Thinking ln(e) = e (ln(e) = 1, because e¹ = e).
• Not recognizing that ln(1) = 0 and ln(e^x) = x.

Explainer

You have already worked with exponential functions like 2^x and 10^x. The number e (approximately 2.71828) looks like an odd choice for a base, but it emerges from a concrete question: what happens if you compound interest not monthly, not daily, but continuously — every instant? Start with $1 at 100% annual interest. Compounded n times per year, the balance after one year is (1 + 1/n)ⁿ. As n grows toward infinity, this expression does not blow up — it converges to a specific constant: e. That is where e comes from, and why it appears throughout every model of continuous growth or decay.

The natural logarithm, written ln(x), is simply the logarithm with base e. Just as log₁₀(1000) = 3 because 10³ = 1000, ln(e³) = 3 because e³ is indeed e³. Because ln and e^x are inverse functions, they undo each other: e^{ln x} = x and ln(e^x) = x. This inverse relationship is the key to solving equations — apply one function to both sides to cancel the other.

Three values are worth internalizing: ln(1) = 0 (because e⁰ = 1), ln(e) = 1 (because e¹ = e), and ln(e^k) = k for any k. A frequent misconception is that ln(e) = e — but the logarithm asks for an exponent, not the base. The answer is always a plain number. Here it is 1.

Continuous growth and decay are modeled by A = A₀ · e^kt. If k > 0 the quantity grows; if k < 0 it decays. To solve for time or rate, take ln of both sides: if A/A₀ = e^kt, then ln(A/A₀) = kt. This is the same algebraic move you practiced with other exponential equations, just using ln in place of log.

The deeper reason e is indispensable — one you will explore in calculus — is that the derivative of e^x is e^x itself. It is the only function that is its own rate of change. This makes e the natural base for any process where the rate of change is proportional to the current value: population growth, radioactive decay, Newton's law of cooling, and more.