Logarithm: Solved Examples

Example 1: Find the value of log3 2187.
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A. 6
B. 8
C. 7
D. 9
Solution: We have log3 2187 = log3 37 = 7 × log3 3 = 7 × 1 = 7 (As log3 3 = 1 by rule)
Hence answer is option C
Example 2: Find the value of x if log4 x = 5.
A. 1024
B. 512
C. 500
D. 2048
Solution: Applying the basic logarithm rules, we have loga x = b ⇒ x = ab
So we have log4 x = 5 ⇒ x = 45 ⇒ x = 1024.
Hence answer is option A
Example 3: Solve log√7/log7
A. 1024
B. 512
C. 500
D. 2048
We have log√7/log7 = log71/2/log7 = 1/2(log7)/log7 = 1/2
Hence answer is option A
Example 4: Solve for ‘x:’ the equation is 2log2x – log2 (x – 2) = 3
A. 6
B. 4
C. 1
D. 2
Solution: We have 2log2x – log2 (x – 2) = 3
⇒ log2x2 – log2 (x – 2) = 3
⇒ log2(x2/x-2) = 3
⇒ log2(x2/x-2) = 23 = 8
⇒ x2 = 8(x-2)
⇒ x2 - 8x + 16 = 0
⇒ (x-4)2 = 0
⇒ x=4
Hence answer is option B
Example 5: If log 2 = 0.301, then find the number of digits in 242.
A. 12
B. 11
C. 13
D. 14
Solution: To find the number of digits, just find the value of log 242.
We have log 242 = 42 log 2 = 42 × (0.301) = 12.642
Now here the integral part is called the characteristic of logarithm and the fractional part is called the mantissa. The number of digits is always one more than the value of characteristic.
The number of digits in 242 = 12 + 1 = 13
Hence answer is option C
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