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Logarithm

Theory and Formula



Questions

Q1. If logy x . logx y = x and x2 + y = 3 , find the value of x and y .

(a) (1,1) (b) (1,2) (c) (1,3) (d) (2,4)

Solution.
(logx/logy) . (logy/logx) = x

x = 1

(1)2 + y = 3

1 + y = 3

y = 2
Solution

Q2. If log16(log3x) =
1/2
, find x

Solution.
log16(log3x) =
1/2
log1616

log16(log3x) = log16(16)1/2

log3x = 4   ⇒   log3x = 4log33

log3x = log3(3)4   ⇒    x = 81
Solution

Q3. log8(log5x) = 1/3, find x.

(a) 15 (b) 26 (c) 25 (d) 10

Solution.
log8(log5x) = 1/3log88

log8(log5x) = log881/3

log8(log5x) = log82

log5x = 2log55

log5x = log552

x = 25
Solution

Q4. log64 (log4 √x32) =
2/3
, find x.

Solution.
log64 (log4 √x32) = log64(64)2/3

log4 (√x32) = 42

log4 (√x32) = 16log44

log4 (√x32) = log4 416

√x32 = 416   ⇒    n16 = 416   ⇒    n = 4
Solution

Q5. Find the value of log100.001

Solution.
log100.001 = x , so that 10x = 0.001 = 10-3   ⇒    x = -3
Solution

Q6. Evaluate 2log 3 + log(1/9)

(a) 3 (b) 0 (c) 1 (d) 2

Solution.
= log 32 + log 1 - log 9

= log 9 + log 1 - log 9

= log 1

= 0
Solution

Q7.
log97/log617
-
log37/log√617
= x , find x .

(a) 0 (b) 1 (c) 2 (d) 3

Solution.
log37/log617
-
log37/log617
= 0

{log97 =
log 7/log 9
=
log 7/2log 3
=
1/2
log37 =
log37/2
}
Solution

Q8. Find a, if log2 a = 4 + log24

(a) 30 (b) 64 (c) 60 (d) 45

Solution.
log2 a - log2 4 = 4 log2 2

log2 (a/4) = log2 (24)

a/4 = 16

a = 64
Solution

Q9. Which one is correct .
(a) Logarithm is not defined for negative number.
(b) log 1 = 1
(c) log22 = 2
(d) lognn = (log n/log m)

Solution.
(a) Logarithm is not defined for negative number.
Solution

Q10. log√(27+10√2) = log(5+x), find x.

(a) 4 (b) 1 (c) √2 (d) √3

Solution.
log√[5 + √2]2 = log(5 + x)

5 + √2 = 5 + x

x = √2
Solution