Logarithmic functions are the inverses of exponential functions. inverse of y = a^{x} is x = a^{y} or y = log_{a} x

## What is an exponential function?

An exponential function is a mathematical function in form f(x) = a^{x} where x is a variable and a is a constant also a is called the base of the function and x is called the exponent of the function

When a is Euler’s number (a = e) then called as a natural exponential function, where e = 2.71828…

## Formula of exponential functions

An exponential function is represented as f(x) = a^{x} where

x is a real number (varying from -∞ to ∞)

a is a constant, grater than 0 and not equal to 1 (a > 0 & a ≠ 1)

## Properties of exponential functions

- When x = 0 then f(x) = a
^{0}= 1 that is f(x) always pass through (0 ,1) - When 0 < a < 1 then f(x) is decreasing to minus infinity, When 1 < a < ∞ then f(x) is increasing to infinity
- f(x) = f(y) then x = y (Let f(x) = a
^{x}& f(y) = a^{y}, then a^{x}= a^{y}hence x = y) (one-one function) - f(x) = a
^{x}is never intersect on x axis

### Why base of an exponential functions cannot be negative ( why f(x) = (-2)^{x} is not an exponential function )

When f(x) = (-2)^{x}, when x is even then f(x) is increasing (positive value) and x is odd then f(x) is decreasing (negative value), so f(x) is increasing and decreasing but exponential functions are either increase or decrease. So f(x) = (-2)^{x} is not an exponential function or base of an exponential function cannot be negative

### Base of the exponential function cannot be 0 & 1? ( why f(x) = 0^{x} & f(x) = 1^{x} is not an exponential function? )

When f(x) = 0^{x} then f(x) is always equal to 0 its independent of the value of x or value of f(x) is 0, neither increasing nor decreasing. Then f(x) = f(y) = 0 but x not equal to y exponential functions are defined as one – one function

When f(x) = 1^{x} we have the same condition of f(x) = 0^{x}. Where f(x) = f(y) = 1 but x is not equal to y

## What is a logarithmic function?

A logarithmic function is a mathematical function in form f(x) = log_{a} x where x is the variable and a is the base of the logarithmic function, where a is called the base of the function and x is the exponent of the function

Logarithmic functions are the inverses of the exponential functions,

That is y = a^{x} can be rewritten as x = log_{a} y

Exponential function can be expressed in logarithmic form. also, all logarithmic functions can be rewritten as an exponential form

When a is Euler’s number (a = e) then called a natural logarithmic function, where e = 2.71828…

## Formula of logarithmic function

An logarithmic function is represented as f(x) = log_{a} x where

x is a real number (varying from 0 to ∞ (not include 0))

a is a constant, greater than 0 and not equal to 1 (a > 0 & a ≠ 1)

## Properties of logarithmic function

- When x = 1 then f(x) = log
_{a}1 = 0 that is f(x) always pass through (1 ,0) - When 0 < a < 1 then f(x) is decreasing to minus infinity, When 1 < a < ∞ then f(x) is increasing to infinity
- f(x) = f(y) then x = y (Let f(x) = log
_{a}x & f(y) = log_{a}y , then log_{a}x = log_{a}y hence x = y) (one-one function) - f(x) = log
_{a}x is never intersect on y axis

## Why logarithm of negative numbers are undefined?

We know logarithmic functions are inverse of the exponential functions f(x) = (-a)^{x} is not defined so inverse functions are not defined

## Why base of a logarithmic function cannot be 0, 1 and negative?

f(x) = 1^{x} is not defined so inverse function of f(x), f(x) = log_{1} x is not defined

f(x) = 0^{x} is not defined so inverse function of f(x), f(x) = log_{0} x is not defined

## Properties of exponents and logarithms

Properties of Exponent | Properties of Logarithm | |

Product Property | a^{x} × b^{x} = a^{x + y} | log_{a} (xy) = log_{a} x + log_{a} y |

Quotient Property | a^{x}/a^{y} = a^{x – y} | log_{a} (x/y) = log_{a} x – log_{a} y |

Power Property | (a^{x})^{y} = a^{x×y} = a^{xy} | log_{a} x^{y} = y log_{a} x |

Inverse Property | a^{loga x} = x | log_{a} a^{x} = x |