Exponential and Logarithmic functions

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) = a⁰ = 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) = log₁ x is not defined

f(x) = 0^x is not defined so inverse function of f(x), f(x) = log₀ 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^{log_a x} = x	log_a a^x = x