Logarithm formula Class 9

Logarithms Class 9

         If am = n , then exponent 'm' is called the logarithm of number 'n' to base a.

                          

                                     am = n is called the exponent form,

          where a is the base,  m is the exponent and n is the number. Then, 

${\log }_{a}n\; =\; m\; is\; called\; the\; logarithm\; form\; and\; is\; read\; as\; log\; n\; to\; the\; base\; a.$

$Thus$

${\log }_{a}n\; =\; m\; if$am = n 

Exercise 

1. Express the following in logarithmic form.

i) 42 = 16

 

$\mathrm{lo}{g}_4\mathrm{16}=2$

ii) 73 = 343

$lo{g}_7\mathrm{343}=3$

iii) 3-2 = 1/9 

$\mathrm{lo}{g}_3\mathrm{1/9}=\mathrm{-2}$

iv) 60 = 1

$lo{g}_{6}1=0$

v) 361/2 = 6

$lo{g}_{\mathrm{36}}6=\mathrm{1/2}$

vi) 10-3 = 0.001

$lo{g}_{\mathrm{10}}\mathrm{0.001}=\mathrm{-3}$

vii) 25 = 1252/3

$lo{g}_{\mathrm{125}}25\; =\mathrm{2/3}$

2. Change the following into exponent form.

i) log381 =4 Ans : 34 = 81$𝑙𝑜{𝑔}_{𝑎}𝑁=𝑥$

$lo{g}_{\mathrm{125}}25\; =\mathrm{2/3}$$lo{g}_{\mathrm{125}}25\; =\mathrm{2/3}$

ii) log525 =2 Ans : 52 = 25

iii) log2781 =4/3 Ans : 274/3 = 81

$o{g}_{\mathrm{125}}25\; =\mathrm{2/3}$

 

iv) log21/4 =-2 Ans : 2-2 = 1/4

v) log51/125 =-3 Ans : 5-3 = 1/125

vi) log100.01 =-2 Ans : 10-2 = 0.01

3. Find the value of :

i) log464

let it is equal to be x

log464 = x

                  4x = 64

$4x=$43

base are same so, x = 3

ii) log216

let it is equal to be x

log216 = x

                  2x = 16

$2x=$24

base are same so, x = 4

iii) log31/9

let it is equal to be x

log31/9 = x

                3x = 1/9

$3x=$1/32

$3x=$3-2

base are same so, x = -2

iv) log1/216

let it is equal to be x

log1/216 = x

               1/2x = 16

$1/2x=$24

$1/2x=\; 1/$2-4

base are same so, x = - 4

4. Find the value of x in the following

i) logx16 = 4

so, in expansion form it is written as

               x4 = 16

x4 = 24

power are same so, x = 2

ii) logx1/27 = 3

x3 = 1/27

x3 = (1/3)3

power are same so, x = 1/3

iii) log√5( x + 8) = 4

                  √ 54= x + 8

25 = x + 8

x = 25 - 8

x = 17

power are same so, x = 17

5 . If log10a = m express 103m in the term of a

             here           10m = a

                   103m = (10m)3

= a3 ------------------------ (10m = a)

1.   Express as a single logarithm.

 1) log 3 + log 4 + log 5  = log 3 * 4 * 5  = log 60

 2) log 2 - 1/3 log 125 + 1/4 log 81 = log 2 - log 1251/3 + 1/4 log 811/4

                                                            = log 2 - log (53)1/3 + log (34)1/4

   

                                                            = log 2 - log 5 + log 3

                                                      

                                                            = log 2 * 3 /5

        

                                                            = log 6/5

3) 1/2 log 36 + 2 log 5 - 3 log 2 = log 361/2 + log 52 - log 23

                                                    = log (62)1/2 + log 25 - log 8

   

                                                    = log 6 + log 25 - log 8

                                                     

                                                    = log 6 * 25 / 8

                                                    = log 150 / 8

                                         

                                                    = log  75/4

4)  1 + log 4 - log 5 = log 10 + log 4 - log 5 .........................( Log1010 = 1)

                                = log 10 * 4 / 5

                                = log 40 / 5

                                = log 8

5) 3 - 2 log 5 + 3 log 2 = log 1000 - log 52+ log 23 .........................( Log101000 = 3)

                                     = log 1000 - log 25 + log 8

                                     = log 1000 * 8 /25

                                     = log 8000/25

= log 320

6) 2/3 log 8 + 6 log 3√2 - 1/2 log 1/9

= log 82/3+ log (3√2)6 - log 1/91/2

= log (23)1/3+ log 8 - log ( 3-2)1/2

= log (23)1/3+ log 8 + log ( 32)1/2

= log 2 + log 8 + log 3

=log 2 * 8 * 3

= log 48

2. Evalute the following 

1. log 8 +2 log 5 - 1/3 log 8

= log 8 + log 52 -  log 81/3

= log 8 + log 25 -  (log 23)1/3

= log 8 + log 25 - log 2

= log 8 * 25 /2

= log 200/2

= log 100 .........................( Log10100 = 2)

= 2

2. 1 + 3 log 5 + 3 log 2 

= log 10 + log 53 + log 23          .........................( Log1010 = 1)

= log 10 + log 125 + log 8

= log 10 * 125 *8

= log 10000    .........................( Log1010000 = 4)

= 4

3. log 81/8 + 2 log 2/3 - 3 log 3/2 + log 3/4

= log 81/8 + log (2/3)2 - log (3/2)3+ log 3/4

= log 81/8 + log 4/9 - log 27/8 + log 3/4

= log 81/8 * 4/9 *3/4 ÷ log 27/8

= log 81/8 * 4/9 * 3/4 * 8/27

= log 1

= 0

3) Solve for x

1) log (x + 6 ) + log (x - 6) = 2 log 8

    log (x + 6 ) ( x - 6) = log 82

    log (x + 6 ) ( x - 6) = log 64

  

    log x2 - 36  = log 64

    so,

    x2 - 36  =  64

    x2  =  64 + 36

     x2 =  100

     x = 10

2) log ( 2x - 3 ) + log 3 = log ( 2x + 7 )

    log 3 * ( 2x - 3 ) = Log 2x + 7

    log 6x - 9 = log 2x + 7

 so, 

         6x - 9 = 2x + 7 

         6x - 2x = 7 + 9 

                4x = 16

                  x = 16 / 4

                  x = 4

3)  log ( 3x + 5 ) - log ( x - 3 ) = 1

      log ( 3x + 5 ) - log ( x - 3 ) = log 10

      log  ( 3x + 5 ) / log ( x - 3) = log 10

so,

     (3x + 5 ) / ( x - 3 ) = 10

      3x + 5 = 10 ( x - 3 )

      3x + 5 = 10x - 30

         35 = 7 x

           x = 35 / 7

           x = 5

 Olympiad for Class 8