# Logarithmic Functions

**Logarithmic Functions**

The representation of the Logarithmic functions as shown;

f(x)=log_{b}(x)** **

Here b signifies the base of the function.

Depending upon the base the function can be a decreasing(value of b lies between 0 to 1) function or an increasing(value b is greater than 1 ) function. Logarithmic functions are also the inverse of exponential functions.

#### Common Logarithmic Function

The logarithmic function with base 10 is called the common logarithmic function and it is denoted by log_{10} or simply log.

f(x) = log_{10} x

#### Natural Logarithmic Function

The logarithmic function to the base e is called the natural logarithmic function and it is denoted by log_{e}^{.}

f(x) = log_{e} x

#### Logarithmic Functions Properties

Logarithmic Functions have some of the properties that allow you to simplify the logarithms when the input is in the form of product, quotient or the value taken to the power. Some of the properties are listed below.

**Product Rule**

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

Multiply two numbers with the same base, then add the exponents.

Example : log 30 + log 2 = log 60

**Quotient Rule **

log_{b }m/n = log_{b} m – log_{b }n

Divide two numbers with the same base, subtract the exponents.

Example : log_{8} 56 – log_{8} 7 = log_{8}(56/7)=log_{8}8 = 1

**Power Rule **

Raise an exponential expression to power and multiply the exponents.

Log_{b }m^{p} =p log_{b} m

Example : log 100^{3} = 3. Log 100 = 3 x 2 = 6

**Zero Exponent Rule **

log_{a} 1 = 0.

**Change of Base Rule**

log_{b }(x) = ln x / ln b or log_{b }(x) = log_{10} x / log_{10} b

**Other Important Rules of Logarithmic Function**

- Log
_{b }b = 1 Example : log_{10}10 = 1 - Log
_{b }b^{x }= x Example : log_{10}10^{x}= x - b
^{log}_{b}^{x}=x. Substitute y=log_{b}x,it becomes b^{y}=x

There are also some of the logarithmic function with fractions. It has a useful property to find the log of a fraction by applying the identities

- ln(ab)= ln(a)+ln(b)
- ln(a
^{x}) = x ln (a)

We also can have logarithmic function with fractional base.

Consider an example,

By the definition, log_{a} b = y becomes a^{y} = b

( $\frac{4}{9}$49 )^{y} = $\frac{27}{8}$278

( $\frac{2^2}{3^2}$2^{2}3^{2} )^{y} = $\frac{3^3}{2^3}$3^{3}2^{3}

(⅔)^{2y} = (3/2)^{3}