The natural log calculator determines the logarithm to the base of a famous mathematical constant, e, an irrational number with an approximate value of e = 2.71828. But, what is the natural logarithm, ln x, of a given number x? This is the power the number e has to be raised to in order to result in a given number x.

How to use the natural logarithm calculator

Like all other logarithms, the natural logarithm of x returns the power, or exponent, to which a given base e must be raised to yield back the number x. It is easier to understand this notion when the base is an integer, for example 2 or 3:

log2 16 = 4 since 24 = 16
log3 81 = 4 since 34 = 81.

In the case of the natural logarithm, this is somewhat less intuitive, because it's base, e, is not an integer. But, since the value of e is between values of 2 and 3, we understand that e4 has to be somewhere between 24 = 16 and 34 = 81.

It turns out that e4 = 54.498. This equality can be stated in terms of the natural logarithm in the following way, which you can check by using the ln calculator:

ln 54.498 = 4.

Here are some examples of natural logarithm:

  • ln 1 = 0 since e0 = 1
  • ln 10 = 2.3026 since e2.3026 = 10
  • ln 20 = 2.996 since e2.996 = 20
  • ln 50 = 3.912 since e3.912 = 50
  • ln 100 = 4.605 since e4.605 = 100.

Check out the correctness of the above results by using our natural log calculator and the exponent calculator. In order to get a better grasp of the interplay between the exponential and logarithm functions, you may want to check out our exponential and logarithm calculators.

Other ways to denote the natural logarithm

One way of denoting the natural logarithm is loge. This is the same as, say, when we write the logarithm to the base two as log2.

But, a vastly more common way to write the natural logarithm is ln, which is an abbreviation of the Latin expression logarithmus naturalis, the name the natural logarithm was given when the Latin language was still the lingua franca of science.

There is also a third way of writing the natural logarithm: log. This notation is somewhat problematic though, as it is often mistaken for the logarithm to the base 10, although this is the syntax many software implementations of natural logarithm use, so be careful!

What's so natural about the natural logarithm?

Why exactly does ln x deserve to be called natural? Maybe the most important property of the natural logarithm is that it is the inverse function of the exponential function ex, the only function whose rate of change, or the derivative, is precisely again itself: (ex)' = ex.

In simpler words, the exponential function ex governs its own rate of change, which in some sense makes it self-sufficient and not dependent on any other function to define the way it changes. Exactly this property makes both ex and its inverse function ln x the natural choice when describing many real-world phenomena.

The natural log graph

Another interesting property of the natural logarithm is the way it changes its values as the argument x increases. One way of expressing this is by stating that the derivative of the natural logarithm is inversely proportional to its value x, which can be written as (ln x)' = 1/x.

You can also observe this property by looking at the graph of the natural logarithm. Although it increases as the value of x increases, the rate of growth is getting smaller and smaller as x approaches higher and higher values:

Natural log graph

Natural log graph
Source: Wikipedia Attribution: Elmextube (Public domain)

Where did the number e come from

At first glance, you wouldn't say that the number e bears any importance to the endeavors of humans or nature. But this is not so! It happens that this number is one of the most important constants in mathematics, so much so that it deserves a proper name. It is called either Euler's number or Napier's constant, depending on whom of these two great mathematicians one wants to attribute its discovery.

Nevertheless, it seems that Leonhard Euler (1707 - 1783) did get more credit, based on the fact that the very letter Euler used to denote this constant is how we denoted it today. Although Euler was the first to calculate e to a significant number of decimal places, he wasn't the first one to discover it: read more about it in the next paragraph.

Leonhard Euler

Leonhard Euler
Source: Wikipedia Attribution: Jakob Emanuel Handmann (Public domain)

How to make e amount of money

Swiss mathematician Jacob Bernoulli (1654 - 1705) stumbled upon the existence of the number e when tackling the problem of compound interest. Let us first understand the idea behind simple interest. For a given initial sum of money, let's say 1 USD, you want to know how much will you have after one year has passed if an interest of 100% is credited only once, and at the end of the year? The answer is easy: 2 USD.

Jacob Bernoulli

Jacob Bernoulli
Source: Wikipedia Attribution: Niklaus Bernoulli (1662-1716) (Public domain)

But, with compound interest, things get a bit more complicated. For example, if the same interest of 100% is now split into two equal parts of 50% and credited twice, 50% at the end of first six months, and another 50% at the end of the year, then the final yield is obtained by using the formula

1*(1 + 1/2)2 = 2.25 USD.

Furthermore, if the interest of 100% is split in weekly lumps, you would have the final yield of

1*(1 + 1/52)52 = 2.692 USD.

Bernoulli asked a simple question: what would happen if compounding was continuous? In other words, what would be the final yield if the interest rate of 100% is split into infinite parts, each credited at the end of an infinitely short period of time? To see in more detail how continuous and other types of compounding work, you may want to check out our investment calculator.

This problem of continuous compound interest, stated mathematically, boils down to the challenge of calculating the limit of (1 + 1/n)n as n approaches infinity. It turns out that the result is precisely the number e! Moreover, the above expression may be used as a way to define e. To answer the Bernoulli's question: with continuous compounding the initial dollar would yield precisely e = 2.718281828 USD at the end of the year!

The real-world importance of ln 2 and other natural logarithms

The easiest natural logarithms to calculate are:

ln 1 = 0 since e0 = 1, and
ln e = 1 since e1 = e.

But, presumably the most important natural logarithm is the one that calculates the value of a number between 1 and e, which turns out to be the number 2. Using the natural log calculator we get

ln 2 = 0.6931.

It turns out that ln 2 is also equal to the alternating sum of reciprocals of all natural numbers:

ln 2 = 1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + ....

At first glance, it appears that this number has no particular importance whatsoever. But, ln 2, as obscure as it may seem, appears in some pretty significant and at first glance unrelated real-world problems.

For example, it has its role in the formula for the half-life of radioactively decaying matter, or in the calculation of time needed to double the initial amount of money, if a fixed rate is applied over a given time.

So, if you have 1,000 USD in your bank account, and the bank provides an r = 7% per annum interest rate, you may want to ask yourself how long it will take to double my initial amount? Well, this is where ln 2 comes into play: the natural logarithm formula that calculates the time needed is (100 * ln 2)/r, which can be simplified to the approximate value of 70/r. Besides 70, the number 72 is used even more often for this purpose, because it has more divisors: check out the rule of 72 calculator!

Therefore, in the case of r = 7%, you would get 70/7 = 10 years as an approximate time needed for your initial sum of money to double. Similarly, one can get analogous formulas for the time needed for the initial quantity to triple, quadruple, or n-tuple, given a fixed rate of growth over time.

Other applications of the natural logarithm

From the previous paragraph, we may conclude that natural logarithms occur in every process with a period-related, constant growth or decay of some quantifiable phenomenon.

Apart from the already mentioned examples of radioactive decay and the problem of yield with a fixed interest rate, natural logarithms appear when calculating the growth and decay of any bacterial, animal and plant population, the decay rates of a charged capacitor, or the temperature change of an object.

References

Miroslav Jerkovic, PhD