Logarithm: Solved Examples

Example 1: Find the value of log3 2187.

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.

Solution: Applying the basic logarithm rules, we have loga x = b ⇒ x = a^b
So we have log₄ x = 5 ⇒ x = 4⁵ ⇒ x = 1024.
Hence answer is option A

Example 3: Solve log√7/log7

We have log√7/log7 = log7^1/2/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

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.

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