Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, log_e x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), log_e(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

The natural logarithm of x is the power to which e would have to be raised to equal x. For example, ln 7.5 is 2.0149..., because e^2.0149... = 7.5. The natural logarithm of e itself, ln e, is 1, because e¹ = e, while the natural logarithm of 1 is 0, since e⁰ = 1.

The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a (with the area being negative when 0 e x both mean the natural logarithm of x. In many areas like mathematics and some programming languages, log x without a base specified can also mean the natural logarithm. However, in other fields such as chemistry, log x might mean the common (base 10) logarithm, or in computer science, it could mean the binary (base 2) logarithm, especially when talking about time complexity.

The general way to show the logarithm of a number x with base b is log_b x. For example, the logarithm of 8 with base 2 is log₂ 8 = 3.

Definitions

The natural logarithm is a special kind of logarithm that uses a number called "e" as its base. The number e is approximately 2.71828 and is very important in mathematics.

One way to understand the natural logarithm is that it is the opposite, or inverse, of the exponential function with base e. This means that if you raise e to the power of the natural logarithm of a number, you get back the original number. For example, e raised to the natural logarithm of x equals x.

Another way to define the natural logarithm is by looking at the area under a special curve called a hyperbola. The hyperbola has the equation y = 1/x. The natural logarithm of a number a is the area under this curve between x = 1 and x = a. If a is smaller than 1, the area will be negative, and so will the logarithm.

Properties

The natural logarithm has some interesting properties. First, the natural logarithm of 1 is 0, and the natural logarithm of the special number e (which is about 2.718) is 1. One useful property is that the logarithm of a product (like multiplying two numbers) is the same as adding their individual logarithms. Similarly, the logarithm of a division (like dividing two numbers) is the difference of their logarithms.

There are also rules for dealing with powers and roots in logarithms. For example, the logarithm of a number raised to a power is the same as multiplying that power by the logarithm of the number itself. These properties help make calculations with logarithms easier and are important in many areas of mathematics.

Derivative

The derivative of the natural logarithm is a way to measure how the logarithm changes as its input changes. For the natural logarithm of a positive number, the derivative is simply 1 divided by that number. This means if you have the natural logarithm of x, the rate of change at any point x is 1/x.

There are different ways to understand why this is true, depending on how the natural logarithm is first defined. If we think of the natural logarithm as an area under a curve, the derivative comes directly from a basic rule of calculus. If we think of it as the opposite of the exponential function, we can use properties of logarithms and limits to show that the derivative is 1/x. This simple result is very useful in many areas of mathematics.

Series

The natural logarithm is a special kind of math operation related to a number and a constant called "e" (about 2.718). It’s often written as "ln". Unlike some other math functions, it doesn’t work at zero.

For numbers close to 1, we can use a special pattern called a "series" to estimate the natural logarithm. Imagine you have a number just a little bigger or smaller than 1 — say 1.1 or 0.9. For these, the pattern helps us find an approximate value.

One famous pattern is called the "Mercator series". It looks like this: for a number very close to 1, you can add up simple pieces to get close to the true value. This was a clever trick used before calculators existed!

Continued fractions

While simple continued fractions aren't available for the natural logarithm, several generalized continued fractions exist. These can help compute natural logarithms quickly, especially for numbers close to 1. By breaking down larger numbers into smaller parts, we can calculate their natural logarithms efficiently.

For instance, the natural logarithm of 2 can be found by expressing 2 as a combination of simpler numbers and using these continued fractions. Similar methods work for finding the natural logarithm of larger numbers like 10.

Complex logarithms

Main article: Complex logarithm

The natural logarithm can also be used with complex numbers. When we use the number e as the base, we can find logarithms for complex numbers, but there are some special challenges. For example, the logarithm isn’t always the same value because adding multiples of 2_iπ_ changes the result. This means the complex logarithm has many possible values, but we can choose a main or “principal” value to work with more easily.