Indian Navy -Logarithm Questions for Practice

                                       Indian Navy Exam Paper

logarithm and number series questions for practice

Logarithm:

If a is a positive real number, other than 1 and am = x, then we write:
m = log_ax and we say that the value of log x to the base a is m.

Examples:

(i). 10³ 1000 log₁₀ 1000 = 3.

(ii). 3⁴ = 81 log₃ 81 = 4.

(iii). 2^-3  ,1/8 log₂ 1/8 = -3.

(iv). (.1)² = .01 log(.₁) .01 = 2.

Properties of Logarithms:

1. log_a (xy) = log_a x + log_a y

2. log_a (x/y)= log_a x - log_a y

3. log_x x = 1

4. loga 1 = 0

5. log_a (xⁿ) = n(log_a x)

6. log_a x = 1/log_x a

7. log_a x = log_b x/log_b a = log x ./ log a

Common Logarithms:

Logarithms to the base 10 are known as common logarithms.

The logarithm of a number contains two parts, namely 'characteristic' and 'mantissa'.

Characteristic: The internal part of the logarithm of a number is called its characteristic.

Case I: When the number is greater than 1.

In this case, the characteristic is one less than the number of digits in the left of the decimal point in the given number.

Case II: When the number is less than 1.

In this case, the characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and it is negative.

Instead of -1, -2 etc. we write 1 (one bar),2 (two bar),etc.

Examples:-
 

Mantissa:

The decimal part of the logarithm of a number is known is its mantissa. For mantissa, we look through log table.

1. Which of the following statements is not correct?

A. log₁₀ 10 = 1
B. log (2 + 3) = log (2 x 3)
C. log₁₀ 1 = 0
D. log (1 + 2 + 3) = log 1 + log 2 + log 3

Answer: Option B

Explanation:

(a) Since log_a a = 1, so log₁₀ 10 = 1.

(b) log (2 + 3) = log 5 and log (2 x 3) = log 6 = log 2 + log 3

log (2 + 3) log (2 x 3)

(c) Since loga 1 = 0, so log10 1 = 0.

(d) log (1 + 2 + 3) = log 6 = log (1 x 2 x 3) = log 1 + log 2 + log 3.

So, (b) is incorrect.

2    If log 2 = 0.3010 and log 3 = 0.4771, the value of log5 512 is:

A. 2.870 

B. 2.967

C. 3.876 

D. 3.912

Answer: Option C

Explanation:

3    If log 27 = 1.431, then the value of log 9 is:

Explanation:

log 27 = 1.431

log (33 ) = 1.431

3 log 3 = 1.431

log 3 = 0.477

log 9 = log(32 ) = 2 log 3 = (2 x 0.477) = 0.954.

4    If log₁₀ 5 + log10 (5x + 1) = log10 (x + 5) + 1, then x is equal to:

log10 5 + log10 (5x + 1) = log10 (x + 5) + log10 10

log10 [5 (5x + 1)] = log10 [10(x + 5)]

5(5x + 1) = 10(x + 5)

5x + 1 = 2x + 10

3x = 9

x = 3

5    The value of log₂ 16 is:

Answer: Option B

Explanation:

Let log2 16 = n.

Then, 2n = 16 = 24 n = 4.

log2 16 = 4.

Directions : In each of the following questions, a number series is given with one term missing. Choose the correct alternative that will continue the same pattern and fill in the blank spaces.

1. 1, 4, 9, 16, 25, (...)

(a) 35

(b) 36 (Ans)

(c) 48

(d) 49

Ans : The numbers are 1², 2², 3², 4², 5².

∴ Missing number = 6² = 36.

2. 20, 19, 17, (...),10, 5,

(a) 12

(b) 13

(c) 14 (Ans)

(d) 15

Ans :  The Pattern is -1, -2, ...

∴ Missing number = 17 - 3 = 14.

3. 2, 3, 5, 7, 11, (...),17

(a) 12

(b) 13 (Ans)

(c) 14

(d) 15

Ans :  Clearly, the given series consists of prime numbers starting from 2. The prime number after 11 is 13. So, 13 is the missing number.

4. 6, 11, 21, 36, 56, (...)

(a) 42

(b) 51

(c) 81 (Ans)

(d) 91

Ans :  The pattern is + 5, + 10, + 15, + 20, .....

∴ Missing number = 56 + 25 = 81.

5. 1, 6, 13, 22, 33, (...)

(a) 44

(b) 45

(c) 46 (Ans)

(d) 47

Ans :  The pattern is + 5, + 7, + 9, + 11, ....

∴ Missing number = 33 + 13 = 46.

6. 3, 9, 27, 81, (...)

(a) 324

(b) 243 (Ans)

(c) 210

(d) 162

Ans :  Each term of the given series is obtained by multiplying its preceding term by 3.

∴ Missing number = 81 x 3 = 243.

7. 1, 9, 17, 33, 49, 73, (...)

(a) 97 (Ans)

(b) 98

(c) 99

(d) 100

Ans :  The pattern is + 8, + 8, + 16, + 16, + 24, ...

∴ Missing number = 73 + 24 = 97.

8. 2, 5, 9, (...),20, 27,

(a) 14 (Ans)

(b) 16

(c) 18

(d) 24

Ans :  The pattern is + 3, + 4, ...

∴ Missing number = 9 + 5 = 14.

9. 5, 9, 17, 29, 45, (...)

(a) 60

(b) 65 (Ans)

(c) 68

(d) 70

Ans :  The pattern is + 4, + 8, + 12, + 16, ...

∴ Missing number = 45 + 20 = 65.

10. 3, 7, 15, 31, 63, (...)

(a) 92

(b) 115

(c) 127 (Ans)

(d) 131

Ans :  Each number in the series is the preceding number multiplied by 2 and then increased by 1.

Thus, (3 x 2) + 1 = 7, (7 x 2) + 1 = 15, (15 x 2) 1 = 31 and so on.

∴ Missing number = (63 x 2) + 1 = 127.