LESSON 5: DEFINITION AND LAWS OF LOGARITHMS

"professor baldeh,image courtesy of workshop pictures"

Objectives

Define a log to any base Convert from logarithmic to index form and vice-versa Deduce the laws of logarithms

Definition

The log of a number is the power to which the base must be raised to give that number.

Laws of logarithms

1. The multiplication law
2. The division law
3. The power law

The multiplication law

Let

                        logbM = x 

and

                        logbN = y

or in index form

                            M = bx  

and

                            N =by

Now

                           MN =bx x by

and

                           MN =bx+y 

or in log form

                      logbMN = x + y
              

Hence

logbMN = logbM + logbN

The division law

Now

                          M/N = bx/by

and

                          M/N = b(x-y)

or in log form

                          logbM/N = x-y

Hence

logbM/N =logbM - logbN

The power law

Now

                        Mn = (bx)n

and

                        Mn = bnx

or in log form

logbMn = n(logbM)

Other special logs

The value of log_b1

Let

                          logb1 = x

then in index form

                             1 =bx

Hence

                           logb1 = 0

therefore

To any base the value of log1 is zero

The value of log_bb

Let

                              logbb = x

then in index form

                              b = bx

Hence

                              logbb = 1

Therefore

The value of the log of a number to the same base is unity

The value of log_b0

Let

                               logbo = x

then in index form

                               0 = bx

Hence

                               logb0 = -infinity

Therefore

To any base the log of zero is minus infinity

The value of log_b(-N)

Let

                                logb(-N) = x

then in index form

                                 -N = bx

Hence

                                 logb(-N) has no real value

Therefore

Only positve numbers have real logarithms

question

Find the value of x in each of the following

                          a. logx9 = 2

solution

in index form

                             x2 = 9

and

                             x =√9

therefore

                             x = 3  ans

                         b. log7x = 0

solution

in index form

                           x = 70

therefore

                           x = 1   ans

Summary

The above laws and thier deduction is very important for all students wishing to do courses in engineering

Assignment

Practice how to deduce the laws above,and do one example on each law

Reading

ref:mathematics for technicians(national NII)1990

Malackt 15:49, 26 February 2007 (CET)