The Natural Logarithm

The logarithm with base e

The natural logarithm is the logarithm to base e, where e is a fixed number, about 2.718. It is written ln, so ln x means the logarithm of x to base e: the power to which e must be raised to give x. Like $\pi$π, e is irrational: its decimals never end and never repeat. To 5 decimal places, $e\approx 2.71828$e ≈ 2.71828.

A calculator has an ln key beside its log key, and $e^x$eˣ is usually the second function on the ln key, because the two undo each other.

$e\approx 2.718$e ≈ 2.718 lies between 2 and 3, nearer to 3.

Where e comes from: a look ahead

Here is one place e turns up. Put $1 in an account that pays 100% interest a year. Paid once, at the end of the year, the interest doubles the money, so the year ends with $2.

Now pay half the rate, 50%, twice a year. Each half-year multiplies the money by 1.5, so the year ends with $1.5^2=2.25$1.5² = 2.25 dollars. Pay $\frac{1}{12}$1/12 of the rate every month and the year ends with ${\left(1+\frac{1}{12}\right)}^{12}=2.61$(1 + 1/12)¹² = 2.61 dollars, to the nearest cent. Paid n times a year, the dollar becomes ${\left(1+\frac{1}{n}\right)}^n$(1 + 1/n)ⁿ dollars.

Paying more often always helps, but by less and less. Paid every day, the dollar becomes ${\left(1+\frac{1}{365}\right)}^{365}=2.7146$(1 + 1/365)³⁶⁵ = 2.7146 dollars, to 4 decimal places. Paid a million times a year, it becomes 2.71828 dollars, to 5 decimal places. As n grows without limit, ${\left(1+\frac{1}{n}\right)}^n$(1 + 1/n)ⁿ closes in on $e\approx 2.71828$e ≈ 2.71828, the amount the dollar would reach if the interest were added continuously. That is what "growth that is continuous" means, and why e is the natural base for it.

The points show ${\left(1+\frac{1}{n}\right)}^n$(1 + 1/n)ⁿ for n = 1 to 12: 2, 2.25, 2.37, and on to 2.61 at n = 12. They climb by less and less, toward the line $y=e\approx 2.718$y = e ≈ 2.718, and never reach it.

ln undoes e to the power

ln asks what power of e gives a number. For the number $e^x$eˣ the answer is x, because x is the power e was raised to. So $\ln \left(e^x\right)=x$ln(eˣ) = x for every x. For example, $\ln \left(e^3\right)=3$ln(e³) = 3 and $\ln \left(e^{-2}\right)=-2$ln(e^(−2)) = −2.

Raise e to the power x, then take ln: the result is x again.

e to the power undoes ln

The other order works too. ln x is the power of e that gives x, so raising e to that power gives x: $e^{\ln x}=x$e^(ln x) = x, for every positive x. A calculator gives ln 20 = 2.9957, to 4 decimal places, and $e^{2.9957}$e^2.9957 is 20, to that accuracy.

So ln and e to the power are inverse functions: each undoes the other. That is how an equation with e in the exponent is solved. If $e^x=20$eˣ = 20, take ln of both sides: x = ln 20 = 2.996, to 3 decimal places. And if ln x = 3, raise e to the power of both sides: $x=e^3=20.09$x = e³ = 20.09, to 2 decimal places.

Take ln of a positive number x, then raise e to that power: the result is x again.

ln 1 = 0 and ln e = 1

$e^0=1$e⁰ = 1, so the power of e that gives 1 is 0: ln 1 = 0. $e^1=e$e¹ = e, so the power of e that gives e is 1: ln e = 1. These are the base-e cases of two facts true in every base, that the logarithm of 1 is 0 and the logarithm of the base is 1.

For a number between 0 and 1, ln is negative: ln 0.5 = −0.6931, to 4 decimal places, because e must be raised to a negative power to give less than 1. And ln 0 and the ln of a negative number do not exist, because every power of e is positive.

The graphs of $y=e^x$y = eˣ and y = ln x

Because each function undoes the other, the graph of y = ln x is the reflection of the graph of $y=e^x$y = eˣ in the line y = x. The point (0, 1) on $y=e^x$y = eˣ, which says $e^0=1$e⁰ = 1, becomes (1, 0) on y = ln x, which says ln 1 = 0. The point (1, e) becomes (e, 1).

$y=e^x$y = eˣ is defined for every x and gives only positive values. y = ln x is defined only for positive x and gives every value: it runs down alongside the y-axis without touching it, and climbs more and more slowly to the right.

The gold curve is y = ln x, the white curve is $y=e^x$y = eˣ, and the white straight line is y = x. The points (0, 1) and (1, e) on $y=e^x$y = eˣ reflect in y = x to (1, 0) and (e, 1) on y = ln x.

Every law carries over

ln is a logarithm like any other, so the laws of logarithms hold for it unchanged: ln(ab) = ln a + ln b, $\ln \left(a/b\right)=\ln a-\ln b$ln(a / b) = ln a − ln b, and $\ln \left(a^n\right)=n\ln a$ln(aⁿ) = n ln a.

For example, ln 2 = 0.69315 and ln 3 = 1.09861, to 5 decimal places, so ln 6 = ln 2 + ln 3 = 1.79176, and ln 9 = 2 ln 3 = 2.19722. The change of base rule works with ln too: ${\log }_{2}10=\ln 10/\ln 2=2.30259/0.69315=3.3219$log₂ 10 = ln 10 / ln 2 = 2.30259 / 0.69315 = 3.3219, to 4 decimal places.

A look ahead: in calculus, $e^x$eˣ turns out to be the one exponential function whose gradient at every point equals its own height. That is the deeper reason e is called the natural base.

The usual mistakes

Giving ln e = 0. $e^1=e$e¹ = e, so ln e = 1. It is ln 1 that is 0, because $e^0=1$e⁰ = 1.

Giving the number instead of the power. $\ln \left(e^3\right)$ln(e³) is 3, the power, not $e^3$e³, and not e.

Undoing ln with a power of 10. The partner of ln is e to the power: if ln x = 2, then $x=e^2=7.39$x = e² = 7.39, to 2 decimal places, not $10^2=100$10² = 100. The partner of $10^x$10ˣ is the common logarithm, log.

Splitting the ln of a sum. ln(a + b) is not ln a + ln b; the laws are about products, quotients and powers.

A model with ln x in it

The next problem uses a model of the form y = a + b ln x. Two readings fix its two numbers a and b. At x = 1, ln 1 = 0, so the model gives y = a + 0 = a: the reading at x = 1 is a itself. A second reading, at another value of x, then gives an equation with only b unknown.

Such a model keeps rising for as long as x grows, but by less and less each time, because ln x rises more and more slowly: from x = 1 to x = 5, ln x rises by 1.609, and from x = 5 to x = 9 by only 0.588, to 3 decimal places.