# Logarithms: Definition, Types, Rules, and Properties

Another approach to writing exponents in mathematics is with logarithms. The logarithm of a number with a base equals another number. The opposite of exponentiation is a logarithm. For example, if 22=4 then the value of log 4 to the base 2 is 2.

As a result, we can deduce that

Logb x = n  or   bn = x

where b is the logarithmic function’s base.

This statement means “Logarithm of x to the base b is equal to n.”

On that note, let’s learn the definition of logarithms, types, and the different properties of logarithms with examples for better understanding.

## What are Logarithms?

The power to which a number needs to be raised to obtain certain other values is referred to as a logarithm. It is the most convenient way to express large numbers in the Indian numeral system. A logarithm has a number of significant properties that demonstrate how addition and subtraction logarithms can also be represented as multiplication and division of logarithms.

The definition of “the exponent by which b must be raised to yield a” is “the logarithm of a positive real number, a positive real number not equal to 1[nb 1]], with respect to base b.”

i.e. by= a ⇔logba=y

Where,

The two positive real numbers “a” and “b”

A real number is y.

“a” is referred to as an argument and is located inside the log. “b” is referred to as the base and is located at the bottom of the log.

To put it another way, the logarithm provides the answer to the question, “How many times is a number multiplied to get the other number?”

For example, how many times 3’s are multiplied to get the number 27?

The result of multiplying 3 times by 3 is 27.

The logarithm is therefore 3.

The following is how the logarithm form is written:

Log3 (27) = 3 ….(1)

The base 3 logarithm of 27 is therefore 3.

You can also write the above logarithm form as:

3x3x3 = 27

33 = 27 …..(2)

Equations (1) and (2) are hence equivalent.

## Logarithm Types

The two main categories of logarithms that we typically deal with are

• Common Logarithm
• Natural Logarithm

### Common Logarithm

The base 10 logarithm is another name for the common logarithm. It can be expressed as log10 or just log. For instance, a log is how the common logarithm of 1000 is expressed (1000). The common logarithm specifies the number of times we must multiply by 10 in order to obtain the desired output.

For example, log (100) = 2

The result of multiplying 10 by itself twice is 100.

### Natural Logarithm

The natural logarithm is sometimes known as the base e logarithm. The natural logarithm is denoted by the letters Ln or loge. In this instance, “e,” the Euler’s constant, is roughly comparable to 2.71828. For example, the natural logarithm of 78 is denoted by the symbol ln. The natural logarithm tells us how many times to multiply e to get the desired outcome.

For example, ln (78) = 4.357.

Therefore, the base e logarithm of 78 is 4.357.

## Logarithm Rules and Properties

Logarithmic operations can only be carried out according to specific restrictions. Let’s look at each of these properties separately.

### Product Rule

According to this rule, adding the individual logarithms of two logarithmic values results in the multiplication of those values.

Logb (mn)= logb m + logb n

### Division Rule

The difference between each logarithm is equal to the division of two logarithmic values.

Logb (m/n)= logb m – logb n

### Exponential Rule

According to the exponential rule, the exponent times the logarithm of m’s logarithm is equal to m’s logarithm with a rational exponent.

Logb (mn) = n logb m

### Change of Base Rule

Logb m = loga m/ loga b

### Base Switch Rule

logb (a) = 1 / loga (b)

### Derivative of log

The derivative of f(x), if f(x) = logb(x), is given by;

f'(x) = 1/(x ln(b))

### Integral of Log

∫logb(x)dx = x( logb(x) – 1/ln(b) ) + C