CLASS-9SOME IMPORTANT PROBLEM & SOLUTION OF LOGARITHM

SOME IMPORTANT PROBLEM & SOLUTION OF LOGARITHM -

1)  Solve for x

i)  log₁₀ x – log₁₀ (2x – 1) =  1

=>    log₁₀ x – log₁₀ (2x – 1) =  log₁₀ 10

x

=>    log₁₀ (---------)  =   log₁₀ 10

2x – 1

x

=>   ----------  =  10

2x – 1

=>    x = 20 x - 10

=>   19 x =  10

=>     x =  10/19      (Ans.)

ii)    log₇ (x² + 2x) - log₇ (x + 2) = 3

x² + 2x

Ans.)   log₇ (------------)  =  3

x + 2

x (x + 2)

=>      log₇ {------------}  =  3

(x + 2)

=>      log₇ x = 3

=>       x  = 7³ =  343    (Ans.)

iii)   log₁₀ 8 + log₁₀ (8x + 1) = log₁₀ (x + 8) + 1

Ans.)   log₁₀ 8 + log₁₀ (8x + 1) = log₁₀ (x + 8) + 1

=>      log₁₀ 8 + log₁₀ (8x + 1) = log₁₀ (x + 8) + log₁₀ 10

=>      log₁₀ [8 (8 x + 1)] = log₁₀ [(x + 8) 10]

=>       8 (8x + 1) = (x + 8) 10

=>        64x + 8 = 10x + 80

=>        64x – 10x = 80 – 8

=>         54x  =  72

=>          x = 36/27 = 4/3     (Ans.)

Example 2.) Express y in terms of x

If log₁₀ y + 2 log₁₀ x = 2,

Ans.)    log₁₀ y + 2 log₁₀ x = 2

=>       log₁₀ y + 2 log₁₀ x = 2 log₁₀ 10

=>       log₁₀ y + log₁₀ x² =  log₁₀ 10²

=>       log₁₀ (yx²) = log₁₀ 10²

=>          yx² = 10²

=>           y = 100/x²                  (Ans.)

Example.3) If log 2 = 0.3010 and log 3 = 0.4771, find the values of –

i)  log 12 = log (3 X 4)

=  log 3 + log 4

=  log 3 + log 2²

=  log 3 + 2 log 2

Now we will substitute the value of log 2 & log 3, and we find -

=  0.4771 + (2 X 0.3010)

=  0.4771 + 0.6020

=   1.0791         (Ans.)

ii)  log √144 = log (144)⅟²

= 1/2 log 144

=  1/2 log (16 X 9)

=  1/2 log (2⁴ X 3²)

=  1/2 (log 2⁴ + log 3²)

=  1/2 (4 log 2 + 2 log 3)

Now we will substitute the value of log 2 & log 3, and we find -

=  1/2 {(4 X 0.3010) + (2 X 0.4771)}

=  1/2 (1.204 + 0.9542)

=  (1/2  X 2.1582)

=  1.0791               (Ans.)

iii)  log 3√54 = log (54)⅓

=  1/3 log (27 X 2)

=  1/3 log (3³ X 2)

=  1/3 log 3³ + 1/3 log 2

=   1/3 . 3 log 3 + 1/3 log 2

=   log 3 + 1/3 log 2

Now we will substitute the value of log 2 & log 3, and we find -

=  0.4771 + (1/3 X 0.3010)

=  0.4771 + 0.1003

=   0.5774        (Ans.)