Logarithmic Function

We have observed that y = ax is a monotonic function (either strictly) decreasing or strictly increasing). Hence it is invertible,

                                                  logarithmic-function

So y = ax <=> x = loga y

Where x ε [-∞,∞ ] and y ε [0, ∞]

The inverse exponential function x = loga y is known as logarithmic function. Writing it in conventional form it becomes

y = loga x = f(x), x ε [0,∞]

The inverse exponential function x = loga y is known as logarithmic function. Writing it in conventional form it becomes

y = loga x = f(x), x ε [0, ∞].

Properties of logarithmic Function:

(i)     y = logb x is defined for x > 0, b > 0, b ≠ 1.

(ii)    if logb a = c then a = bc

(iii)    logb 1 = 0

(iv)   logb b = 1

(v)    logb a = 1/loga b

(vi)   logb xy = logb x + logb y

(vii)   logb XY = logb x - logb y

(viii)  logb xm = m logb x

(ix)   logbn x = 1/n logb x

(x)    logb bx = x

(xi)   (b)logbx = x

Illustration: Prove logb a = 1/loga b

Solution:

        Let c = logb a      and   d = loga b

        =>     a = bc        and   b = ad

        =>     a = bc        and   a = b(1/d)

        =>     c = 1/d

        =>     logb a = 1/loga b

Illustration: Prove logb xm = m logb x

Solution:

        Let    c = logb xm         and   d = logb x

        =>     xm = bc              and   x = (b)d

        =>     ((b)d)m = bc

        =>     md = c

        =>     logb xm = m logb x.

AskIITians offers a novel way of teaching where you can prepare for IIT JEE, AIEEE and other engineering examinations for free by sitting at home. You can visit the website askIITians.com to read the study material pertaining to your preparation. Be a part of our online tests and AQAD (A Question A Day) for free and be a winner.

 

Contact askiitians experts to get answers to your queries by filling up the form given below:

We promise that your information will be our little secret. To know more please see our Privacy Policy
We promise that your information will be our little secret. To know more please see our Privacy Policy

OR

Sign Up with Facebook

Signing up with Facebook allows you to connect with friends and classmates already using askIItians. It’s an easier way as well. “Relax, we won’t flood your facebook news feed!”

Related Resources
Composite Functions Part-2

Composite Functions Problem of finding out fog and...

Bounded and Unbounded Function

Bounded and Unbounded Function Let a function be...

Set Theory

Set Theory SET A set is a well-defined collection...

Algebra of Functions

Algebra of Functions Given functions f : D →...

Graphical Representation of a Function Part-1

Graphical Representation of a Function The...

Greatest Integer Function

Greatest Integer Function The function f(x) : R...

Graphical Representation of a Function Part-2

Graphical Representation of a Function...

Polynomial And Rational Function

Polynomial And Rational Function A function of the...

Constant Function and Identity Function

Constant Function and the Identity Function The...

Inequalities

INEQUALITIES The following are some very useful...

Basic Transformations on Graphs

Basic Transformations on Graphs Drawing the graph...

Inverse Function

Inverse Function Let f : X → Y be a function...

Exponential Function

Exponential Function Exponential and Logarithmic...

Signum Function

Signum Function The signum function is defined as...

Explicit and Implicit Functions

Explicit and Implicit Functions If, in a function...

Functions One-One/Many-One/Into/Onto

Functions: one-one/many-one/into/onto Functions...

Absolute Value Function

Absolute Value Function The function defined as:...

Periodic Function

Periodic Function These are the function, whose...

Even And Odd Function

Even And Odd Function A function f(x) : X → Y...

Linear Function

Linear Function When the degree of P(x) and Q(x)...

Composite Functions Part-1

Composite Functions Another useful combination of...

Invertible Function Part-1

Invertible Function Let us define a function y =...

Solved Examples Part-1

Download IIT JEE Solved Examples on Set, Relations...

Increasing or Decreasing Function

Increasing or decreasing function The function f...

Introduction to Functions

Introduction to Functions Definition of Function:...

Cartesian Product

Cartesian Product Let A and B are two non-empty...