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.)