Logarithmic functions are the inverses of exponential functions. inverse of y = ax is x = ay or y = loga x

What is an exponential function?

An exponential function is a mathematical function in form f(x) = ax 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) = ax 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

  1. When x = 0 then f(x) = a0 = 1 that is f(x) always pass through (0 ,1)
  2. When 0 < a < 1 then f(x) is decreasing to minus infinity, When 1 < a < ∞ then f(x) is increasing to infinity
  3. f(x) = f(y) then x = y (Let f(x) = ax & f(y) = ay, then ax = ay hence x = y) (one-one function)
  4. f(x) = ax 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) = 0x & f(x) = 1x is not an exponential function? )

When f(x) = 0x 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) = 1x we have the same condition of f(x) = 0x. 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) = loga 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 = ax can be rewritten as x = loga 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) = loga 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

  1. When x = 1 then f(x) = loga 1 = 0 that is f(x) always pass through (1 ,0)
  2. When 0 < a < 1 then f(x) is decreasing to minus infinity, When 1 < a < ∞ then f(x) is increasing to infinity
  3. f(x) = f(y) then x = y (Let f(x) = loga x & f(y) = loga y , then loga x = loga y hence x = y) (one-one function)
  4. f(x) = loga 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) = 1x is not defined so inverse function of f(x), f(x) = log1 x is not defined

f(x) = 0x is not defined so inverse function of f(x), f(x) = log0 x is not defined

Properties of exponents and logarithms

Properties of ExponentProperties of Logarithm
Product Propertyax × bx = ax + y loga (xy) = loga x + loga y
Quotient Propertyax/ay = ax – yloga (x/y) = loga x – loga y
Power Property(ax)y = ax×y = axyloga xy = y loga x
Inverse Propertyaloga x = xloga ax = x