Logarithm

If a^x = b then log_a b = x.

Laws of Logarithms :-

I. log 1 = 0

II. log_a  a = 1 

III. Product Rule :
The logarithm of a product is the sum of the logarithms of the factors.

log_a xy = log_a x + log_a y

IV. Quotient Rule

The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator

log_a = log_a x – log_a y

V. Power Rule

log_a xⁿ = nlog_a x

VI. Change of Base Rule

where x and y are postive, and a > 0, a ≠ 1

VII.  log_a b*  log_b a = 1.

VIII.  a  ^log_a^N = N.

Q1. Evaluate 2 log ₃ 5 + log ₃ 40 – 3 log ₃ 10 
(1) 0 (2) 1 (3) 10 (4) 3 (5) None of these.

Solution :- 
2 log₃ 5 + log₃ 40 – 3 log₃ 10
= log₃ 5² + log₃ 40 – log₃ 10³
= log₃ 25 + log₃ 40 – log₃ 1000
= log₃ (25 * 40 / 1000)
= log₃ 1
= 0 

Q2. Given that log₂ 3 = 1.585 and log₂ 5 = 2.322 then log₄ 15 is closest to.
(1) 0 (2) 1 (3) 2 (4) 3 (5) None of these.

Solution :- 


 



Q3. which of the following is/are not correct ?


I. log₁₀100 = 10
II. log₃(55) = log₃(165) - log₃ (3)

III. log₅ √6 ÷ log₅ 6 = 1 / √2 
IV. log₅ (1+2+3) = log₅ 1 * log₅2 * log₅ 3

1. Statement I & II are incorrect.
2. Statement I, II & III are incorrect.
3. Statement I & III are incorrect.
4. Statement I, III & IV are incorrect.
5. All are incorrect.  

Solution:-
I. log₁₀100 = log₁₀10 = 2 * log₁₀10² = 2 * 1 = 2. Hence I is incorrect.
II.  log₃(165) - log₃ (3) = log₃(165/3) = log₃(55). Hence II is correct.
III. log₅ √6 ÷ log₅ 6 = log₅ √6 ÷ log₅ (√6)²  = 1 / 2. Hence III is incorrect. 
IV. conceptually incorrect.  Hence answer option 4.


Q4.

If log

x/y

+

log

y/x

= log (x + y), then

1. x + y = 1
2. x - y = 1
3. x = y
4. x & y are even
5. None of these


Solution:-

log	x/y	+	log	y/x	= log (x + y)

log (x*y / y*x) = log (x + y)
log 1 = log (x + y)
hence x + y = 1. So answer option 1.

Q5. Find the number of digits in 5^25.

(1) 15 (2) 16 (3) 17 (4) 18 (5) 19.

Solution: Number of digits = smallest integer greater than log₁₀ of number
number of digits = log₁₀ (5²⁵) = 25 log₁₀ 5 = 25 * 0.6989 = 17.47 = 18
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