# LESSON 5: DEFINITION AND LAWS OF LOGARITHMS

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 logb1

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 logbb

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 logb0

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 logb(-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