logarithms

The Natural Logarithm

Algebra II25 minDifficulty

You should know: logarithms, exponential functions

Overview

The natural logarithm ln(x) = log_e(x) uses base e ≈ 2.71828, Euler's number. It is the logarithm that arises naturally in calculus: d/dx[ln x] = 1/x and the integral of 1/x is ln|x| + C. The natural log solves continuous growth/decay models: if a quantity grows at rate r continuously, it multiplies by e^{rt} in time t.

Intuition

e ≈ 2.718 is the unique number where the exponential e^x has its own derivative (d/dx[e^x] = e^x). Consequently, ln(x) has the beautifully simple derivative 1/x. Continuous compounding: $1 at 100% interest for 1 year = e^1 ≈ $2.718. The half-life formula: if N(t) = N_0 * e^{-kt}, then ln(2)/k = half-life.

Formal Definition

Definition

ln(x) = integral_1^x (1/t) dt for x > 0. Properties: ln(1)=0, ln(e)=1, ln(xy)=ln(x)+ln(y), ln(x/y)=ln(x)-ln(y), ln(x^r)=r*ln(x). Derivative: d/dx[ln(x)] = 1/x. Inverse: e^{ln(x)} = x for x>0; ln(e^x) = x for all x. e = lim_{n->inf}(1+1/n)^n = sum_{n=0}^{inf} 1/n!.

\ln(x) = \int_1^x \frac{1}{t}\,dt

Integral definition

\frac{d}{dx}[\ln x] = \frac{1}{x}

Derivative

e = \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n \approx 2.71828

Euler's number

Notation

Notation	Meaning
$\ln(x)$	Natural logarithm: log base e
$e$	Euler's number ≈ 2.71828, base of natural log

Theorems

Theorem 1: Logarithm Derivative and Integral

d/dx[ln|x|] = 1/x for x not 0. Integral: integral (1/x) dx = ln|x| + C. These make ln the natural antiderivative of 1/x in calculus.

Worked Examples

1
Take natural log of both sides.
$2x = \ln(7)$
2
Divide by 2.
$x = \frac{\ln 7}{2} \approx \frac{1.946}{2} \approx 0.973$

✓ Answer

x = ln(7)/2 ≈ 0.973.

Practice Problems

Easyfill in blank

If a population grows as P(t) = 500*e^{0.03t}, when does it double?

Common Mistakes

Common Mistake

Writing ln(x+y) = ln(x) + ln(y).

The product rule says ln(x*y) = ln(x)+ln(y). There is no simplification for ln(x+y).

Quiz

What is ln(e^5)?

5e^55e1/5

Historical Background

The natural logarithm first appeared implicitly through quadrature of the hyperbola y=1/x. Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa (1647) realized the area under 1/x from 1 to x equals what we now call ln(x). Euler formalized e as the base of the natural logarithm and proved that e = lim_{n->inf}(1+1/n)^n. The notation ln was introduced by Stringham in 1893.

1647
De Sarasa connects hyperbola quadrature to logarithm properties
Alphonse Antonio de Sarasa
1736
Euler formalizes e and the natural logarithm in Mechanica
Leonhard Euler

Summary

ln(x) = log_e(x). e ≈ 2.71828. ln(e) = 1, ln(1) = 0.
Product: ln(xy)=ln(x)+ln(y). Power: ln(x^r)=r*ln(x).
d/dx[ln x] = 1/x. Integral: integral(1/x)dx = ln|x|+C.
Continuous growth: N(t) = N_0*e^{rt}. Doubling time = ln(2)/r.

References

BookLarson, R. Algebra and Trigonometry. 9th ed. Brooks/Cole, 2013.