# Logarithms

A logarithm is defined as the power to which a number must be raised to get some other values. It is the most convenient way to express large numbers.

A logarithm of a number with a base is equal to another number. A logarithm is just the opposite function of exponentiation. For example, if 10^{2} = 100 then log_{10} 100 = 2.

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

**i.e. b ^{y}= a ⇔log_{b}a=y**

Where,

- “a” and “b” are two positive real numbers
- y is a real number
- “a” is called argument, which is inside the log
- “b” is called the base, which is at the bottom of the log.

## Logarithm Types

In most cases, we always deal with two different types of logarithms, namely

- Common Logarithm
- Natural Logarithm

### Common Logarithm

The common logarithm is also called the base 10 logarithms. It is represented as log10 or simply log. For example, the common logarithm of 1000 is written as a log (1000). The common logarithm defines how many times we have to multiply the number 10, to get the required output.

For example, log (100) = 2

If we multiply the number 10 twice, we get the result 100.

### Natural Logarithm

The natural logarithm is called the base e logarithm. The natural logarithm is represented as ln or loge. Here, “e” represents the Euler’s constant which is approximately equal to 2.71828. For example, the natural logarithm of 78 is written as ln 78. The natural logarithm defines how many we have to multiply “e” to get the required output.

For example, ln (78) = 4.357.

Thus, the base e logarithm of 78 is equal to 4.357.

## Logarithm Rules and Properties

There are certain rules based on which logarithmic operations can be performed. The names of these rules are:

- Product rule
- Division rule
- Power rule/Exponential Rule
- Change of base rule
- Base switch rule
- Derivative of log
- Integral of log

### Product Rule

In this rule, the multiplication of two logarithmic values is equal to the addition of their individual logarithms.

**Log**_{b}** (mn)= log**_{b}** m + log**_{b}** n**

For example: log_{3 }( 2y ) = log_{3 }(2) + log_{3 }(y)

### Division Rule

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

**Log**_{b}** (m/n)= log**_{b}** m – log**_{b}** n**

For example, log_{3 }( 2/ y ) = log_{3 }(2) -log_{3 }(y)

### Exponential Rule

In the exponential rule, the logarithm of m with a rational exponent is equal to the exponent times its logarithm.

**Log**_{b}** (m**^{n}**) = n log**_{b}** m**

Example: log_{b}(2^{3}) = 3 log_{b} 2

### Change of Base Rule

**Log**_{b }**m = log**_{a }**m/ log**_{a}** b **

Example: log_{b} 2 = log_{a }2/log_{a} b

### Base Switch Rule

**log _{b }(a) = 1 / log_{a }(b)**

Example: log_{b} 8 = 1/log_{8} b

### Derivative of log

If f (x) = log_{b }(x), then the derivative of f(x) is given by;

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

Example: Given, f (x) = log_{10 }(x)

Then, f'(x) = 1/(x ln(10))

### Integral of Log

**∫log _{b}(x)dx = x( log_{b}(x) – 1/ln(b) ) + C**

Example: ∫ log_{10}(x) dx = x ∙ ( log_{10}(x) – 1 / ln(10) ) + C

### Other Properties

Some other properties of logarithmic functions are:

- Log
_{b}b = 1 - Log
_{b}1 = 0 - Log
_{b}0 = undefined

## Logarithmic Formulas

log_{b}(mn) = log_{b}(m) + log_{b}(n)

log_{b}(m/n) = log_{b} (m) – log_{b} (n)

Log_{b} (xy) = y log_{b}(x)

Log_{b}m√n = log_{b} n/m

m log_{b}(x) + n log_{b}(y) = log_{b}(x^{m}y^{n})

log_{b}(m+n) = log_{b} m + log_{b}(1+nm)

log_{b}(m – n) = log_{b} m + log_{b} (1-n/m)