Logarithms are the inverse of exponents. A logarithm expresses the power to which a base number must be raised to equal some value. There are common, natural, and binary logarithms which use bases of 10, e, and 2 respectively. Logarithms have properties like logarithms of products equaling the sum of logarithms. Questions are provided to test understanding of logarithm concepts and properties through calculations.

1. LOGARITHM

2. Logarithm
What are Logarithms?
• Logarithms are the inverse of exponents.
• A logarithm (or log) is the mathematical expression used to answer the question: How
many times must one “base” number be multiplied by itself to get some other particular
number?
• For instance, how many times must a base of 10 be multiplied by itself to get 1,000? The
answer is 3 (1,000 = 10 × 10 × 10).

3. Logarithm
RELATION BETWEEN EXPONENTIATION AND LOGARITHM:
TYPES OF LOGARITHM:
 Common Logarithm: The logarithm base 10 (that is b = 10) is called the common logarithm and is
commonly used in science and engineering.
 Natural Logarithm: The natural logarithm has the number e (that is b ≈ 2.718) as its base; its use is
widespread in mathematics and physics, because of its simpler integral and derivative.
 Binary Logarithm: The binary logarithm uses base 2 (that is b = 2) and is commonly used in computer
science. Logarithms are examples of concave functions.

4. Logarithm

5. Logarithm
Some Additional Properties of Logarithm:

6. 1. log 2 = 0.3010 and log 3 = 0.4771, the values of log5 512 is
A) 2.875 B) 3.875 C) 4.875 D) 5.875
Logarithm

7. Logarithm
2. If log 64 = 1.8061, then the value of log 16 will be (approx)?
A) 1.9048 B) 1.2040 C) 0.9840 D) 1.4521

8. Logarithm
3. Solve for x:
2log(x + 2) = log(x + 2) + 1
A. 8 B. 6 C. 9 D. 10

9. Logarithm
4. Solve : 49log
7
4
A. 7 B. 14 C. 16 D. 18

10. Logarithm
5. If log 3 = 0.477 and (1000)x = 3, then x equals to ?
A. 0.159 B. 10 C. 0.0477 D. 0.0159

11. Logarithm
6. If 100.3010 = 2, then find the value of log0.125 125. ?
A. -699 / 301 B. 699 / 301 C. 1 D. 2

12. Logarithm
8. log9 (3log2 (1 + log3 (1 + 2log2 x))) = 1/2. Find x.
A. 4 B. 12 C. 1 D. 2

13. Logarithm
9. If log X/(a2 + ab + b2) = log Y/(b2 + bc + c2) = log Z/(c2 + ca + a2), then X a – b .Yb - c. Z c - a =?
A. 0 B. -1 C. 1 D. 2

14. Logarithm
10. If log X = (log Y)/2 = (log Z)/5, then X4.Y3.Z-2 =?
A. 2 B. 10 C. 1 D. 0

15. Logarithm
11. If logn48 = a and logn108 = b. What is the value of logn1296 in terms of a and b?
A. 2(2a + b)/5 B. (a + 3b)/5 C. 4(2a + b)/5 D. 2(a + 3b)/5

16. Logarithm
12. What is the value of X, if
1/(log442/441 X) + 1/(log443/442 X) + 1/(log444/443 X) +…+ 1/(log899/898 X) +1/(log900/899 X) = 2?
A. 2/21 B. 1 C. 7/100 D. 10/7

17. Logarithm
13. Solve for X, log10X + log√10 X + log 3√100 X = 27
A. 1 B. 104 C. 106 10

18. Logarithm
14. If (4.2)x = (0.42)y = 100, then (1/x) - (1/y) =
A. 1 B. 2 C. ½ D. -1

19. Logarithm
15. If x = 1998!, then value of the expression
1
𝑙𝑜𝑔2
𝑥
+
1
𝑙𝑜𝑔3
𝑥
+……+
1
𝑙𝑜𝑔1998
𝑥
equals-
A. -1 B. 0 C. 1 D. 1998

20. Logarithm
16. In which of the following m > n
(A)m = (log2 5)2 & n = log2 20
(B) m = log10 2 & n = log10 3√10
(C) m = log10 5. log10 20 + (log10 2)2 & n = 1
(D) m = log1/2 (1/3) & n = log1/3 (1/16)

21. Logarithm
17.

22. Logarithm
18. If log3[log2 (x2 – 4x – 37)] = 1, where ‘x’ is a natural number, find the value of x.
A. 9 B. 10 C. 7 D. 4

23. Logarithm
19. If 3(X - 2) = 5 and log10 2 = 0.3010, log103= 0.4771, then X =?
A. 1 + (2218/4771)
B. 2 + (2218/4771)
C. 2 + (2218/4771)
D. None of these

24. Logarithm
20. If log30 3 = p and log30 5 = q, then, how is log30 8 expressed in terms of p and q
A. 3 – 3(p + q) B. 5p + 2q – 1 C. p + q -2 D. None of these

25. Logarithm
21. If log8 log3 log2 2+3√x=1/3. Find x
A. 512 B. 289 C. 170 D. None of these

26. Logarithm
22. log3x + logx3 = 17/4. Find the value of x.
A. 34 B. 31/4 C. 34 or 31/4 D. 31/3

27. Logarithm
23. Find the value of log10 10 + log10 102 + ……… + log10 10n
A. n∗(2n+1)
B. n∗(n+1)/2
C. 3n∗(n+1)/2
D. n∗(n2+1)/2

28. Logarithm
24. If log2X + log4X = log0.25√6 and x > 0, then x is
A. 6-1/6
B. 61/6
C. 3-1/3
D. 61/3

29. Logarithm
25. If x, y, z be positive real numbers such that log2x z = 3, log5y z = 6 and log xy z = 2/3
then the value of z is in the form of m/n in lowest form then find value of n – m
A. 7 B. 5 C. 3 D. 9

30. Logarithm
26. If x, y and z are the sides of a right angled triangle, where ‘z’ is the hypotenuse, then find
the value of (1/log x+z y) + (1/log z-x y)
A. 1 B. 2 C. 3 D. 4

31. Logarithm
27. If log1087.5 = 1.9421, then the number of digits in (875)10 is?
A. 30 B. 28 C. 33 D. 27

32. Logarithm
28. If log2 = 0.30103 and log3 = 0.4771, then number of digits in 6485 is?
A. 12 B. 13 C. 14 D. 15

33. Logarithm
29. If no. of zeroes after decimal in (0.15)20 is ab. Find b – a.
(Assume log 2 = 0.3010, log 3 = 0.4771)
A. 3 B. 4 C. 5 D. 6

34. Logarithm
30. Find the number of digits in 8^100 ?
A. 90 B. 91 C. 92 D. 100

35. 31. Find the value of z?
1. 10 + log z x = log x z
2. x= 2
A. Statement 1 ALONE is sufficient, but statement 2 alone is not sufficient to answer the
question asked.
B. Statement 2 ALONE is sufficient, but statement 1 alone is not sufficient to answer the
question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but
NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and
additional data specific to the problem are needed.
Logarithm

36. 32. How the value of log 144 ?
1. log 2 = 0.301
2. log 3= 0.477
A. Statement 1 ALONE is sufficient, but statement 2 alone is not sufficient to answer the
question asked.
B. Statement 2 ALONE is sufficient, but statement 1 alone is not sufficient to answer the
question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but
NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and
additional data specific to the problem are needed.
Logarithm

37. 33. Find the number of digits in 9^98?
1. log2= 0.3010
2. log3= 0.477
A. Statement 1 ALONE is sufficient, but statement 2 alone is not sufficient to answer the
question asked.
B. Statement 2 ALONE is sufficient, but statement 1 alone is not sufficient to answer the
question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but
NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and
additional data specific to the problem are needed.
Logarithm

38. 34. Find the value of K?
1. log81 25 = k log3 625
2. log729 5 = k log3 125
A. Statement 1 ALONE is sufficient, but statement 2 alone is not sufficient to answer the
question asked.
B. Statement 2 ALONE is sufficient, but statement 1 alone is not sufficient to answer the
question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but
NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and
additional data specific to the problem are needed.
Logarithm

39. 35. Find the value of x + y?
1. log x /logy= log 125/ log 343
2. log y/log 16 = log 25/log 49
A. Statement 1 ALONE is sufficient, but statement 2 alone is not sufficient to answer the
question asked.
B. Statement 2 ALONE is sufficient, but statement 1 alone is not sufficient to answer the
question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but
NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and
additional data specific to the problem are needed.
Logarithm

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