Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant e. This number e is special and is about equal to 2.71828. The natural logarithm of x is written as ln x, log_e x, or sometimes just log x. Parentheses can be added for clarity, like ln(x), to make things easier to read.

The natural logarithm of x tells us the power to which we must raise e to get x. For example, ln 7.5 is about 2.0149 because e^2.0149... equals 7.5. The natural logarithm of e itself, ln e, is 1 because e¹ = e. The natural logarithm of 1 is 0 because 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.

In mathematics and some programming languages, when we write log x without saying the base, it often means the natural logarithm. But in other areas like chemistry, log x might mean the common logarithm (base 10). In computer science, it could mean the binary logarithm (base 2), 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 features. The natural logarithm of 1 is 0. The natural logarithm of a special number e (which is about 2.718) is 1.

One useful rule is that the logarithm of a product (like multiplying two numbers) is the same as adding their individual logarithms. In the same way, the logarithm of a division (like dividing two numbers) is the difference of their logarithms.

There are also rules for working 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 shows how the logarithm changes when its input changes. For the natural logarithm of a positive number, the derivative is 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. If we think of the natural logarithm as an area under a curve, the derivative comes from a basic rule of calculus. If we think of it as the opposite of the exponential function, we can use properties of logarithms to show that the derivative is 1/x. This simple result is useful in many areas of mathematics.

Series

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

For numbers close to 1, we can use a 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 lets you add up simple pieces to get close to the true value. People used this before calculators existed!

Continued fractions

Simple continued fractions cannot be used for the natural logarithm, but other types of continued fractions can help. These methods are useful for calculating natural logarithms, especially for numbers close to 1. By splitting larger numbers into smaller parts, we can find their natural logarithms more easily.

For example, the natural logarithm of 2 can be calculated by writing 2 as a mix of simpler numbers and using these special fractions. The same ideas can be applied to find the natural logarithm of bigger numbers like 10.

Complex logarithms

Main article: Complex logarithm

The natural logarithm can work with complex numbers too. When we use the number e as the base, we can find logarithms for these numbers. But there are some tricky parts. For example, the logarithm can have many different values because adding multiples of 2_iπ_ changes the answer. This means the complex logarithm has many possible results, but we can pick a main or “principal” value to make things easier.