# Frank solutions for Class 9 Maths ICSE chapter 10 - Logarithms [Latest edition]

## Chapter 10: Logarithms

Exercise 10.1Exercise 10.2
Exercise 10.1

### Frank solutions for Class 9 Maths ICSE Chapter 10 Logarithms Exercise 10.1

Exercise 10.1 | Q 1.01

Express the following in the logarithmic form:
33 = 27

Exercise 10.1 | Q 1.02

Express the following in the logarithmic form:
54 = 625

Exercise 10.1 | Q 1.03

Express the following in the logarithmic form:
90 = 1

Exercise 10.1 | Q 1.04

Express the following in the logarithmic form:

(1)/(8) = 2-3

Exercise 10.1 | Q 1.05

Express the following in the logarithmic form:
112 = 121

Exercise 10.1 | Q 1.06

Express the following in the logarithmic form:

3-2 = (1)/(9)

Exercise 10.1 | Q 1.07

Express the following in the logarithmic form:
10-4 = 0.0001

Exercise 10.1 | Q 1.08

Express the following in the logarithmic form:
70 = 1

Exercise 10.1 | Q 1.09

Express the following in the logarithmic form:

(1/3)^4 = (1)/(81)

Exercise 10.1 | Q 1.1

Express the following in the logarithmic form:

9-4 = (1)/(6561)

Exercise 10.1 | Q 2.01

Express the following in the exponential form:
log2 128 = 7

Exercise 10.1 | Q 2.02

Express the following in the exponential form:
log3 81 = 4

Exercise 10.1 | Q 2.03

Express the following in the exponential form:
log10 0.001 = -3

Exercise 10.1 | Q 2.04

Express the following in the exponential form:

"log"_2 (1)/(32) = -5

Exercise 10.1 | Q 2.05

Express the following in the exponential form:
logb a = c

Exercise 10.1 | Q 2.06

Express the following in the exponential form:

"log"_2 (1)/(2) = -1

Exercise 10.1 | Q 2.07

Express the following in the exponential form:
log5 a = 3

Exercise 10.1 | Q 2.08

Express the following in the exponential form:
"log"_sqrt(3) 27 = 6

Exercise 10.1 | Q 2.09

Express the following in the exponential form:

"log"_25 sqrt(5) = (1)/(4)

Exercise 10.1 | Q 2.1

Express the following in the exponential form:
q = loga p

Exercise 10.1 | Q 2.11

Express the following in the exponential form:
"log"_sqrt(6) (6sqrt(6)) = 3

Exercise 10.1 | Q 2.12

Express the following in the exponential form:
-2 = log2 0.25

Exercise 10.1 | Q 3.01

Find x in the following when: log x 49 = 2

Exercise 10.1 | Q 3.02

Find x in the following when: log x 125 = 3

Exercise 10.1 | Q 3.03

Find x in the following when: log x 243 = 5

Exercise 10.1 | Q 3.04

Find x in the following when: log 8 x = (2)/(3)

Exercise 10.1 | Q 3.05

Find x in the following when: log 7 x = 3

Exercise 10.1 | Q 3.06

Find x in the following when: log 4 x = -4

Exercise 10.1 | Q 3.07

Find x in the following when: log 2 0.5 = x

Exercise 10.1 | Q 3.08

Find x in the following when: log 3 243 = x

Exercise 10.1 | Q 3.09

Find x in the following when: log 10 0.0001 = x

Exercise 10.1 | Q 3.1

Find x in the following when: log 4 0.0625 = x

Exercise 10.1 | Q 4.01

Find the value of: log 10 1000

Exercise 10.1 | Q 4.02

Find the value of: log 3 81

Exercise 10.1 | Q 4.03

Find the value of: log 5 3125

Exercise 10.1 | Q 4.04

Find the value of: log 2 128

Exercise 10.1 | Q 4.05

Find the value of: "log" _(1/5) 125 = x

Exercise 10.1 | Q 4.06

Find the value of: log 10 0.0001

Exercise 10.1 | Q 4.07

Find the value of: log 5 125

Exercise 10.1 | Q 4.08

Find the value of: log 8 2

Exercise 10.1 | Q 4.09

Find the value of: "log_(1/2)16

Exercise 10.1 | Q 4.1

Find the value of: log 0.0110

Exercise 10.1 | Q 4.11

Find the value of: log 3 81

Exercise 10.1 | Q 4.12

Find the value of: log 5 (1)/(25)

Exercise 10.1 | Q 4.13

Find the value of: log 2 8

Exercise 10.1 | Q 4.14

Find the value of: log a a3

Exercise 10.1 | Q 4.15

Find the value of: log 0.1 10

Exercise 10.1 | Q 4.16

Find the value of: "log_sqrt(3) (3sqrt(3))

Exercise 10.1 | Q 5.1

If log 10 x = a, express the following in terms of x : 10 2a

Exercise 10.1 | Q 5.2

If log 10 x = a, express the following in terms of x: 10 a + 3

Exercise 10.1 | Q 5.3

If log 10 x = a, express the following in terms of x: 10 -a

Exercise 10.1 | Q 5.4

If log 10 x = a, express the following in terms of x: 102a -3

Exercise 10.1 | Q 6.1

If log 10 m = n, express the following in terms of m: 10 n -1

Exercise 10.1 | Q 6.2

If log 10 m = n, express the following in terms of m: 10 2n+1

Exercise 10.1 | Q 6.3

If log 10 m = n, express the following in terms of m: 10 -3n

Exercise 10.1 | Q 7.1

If log 10 x = p, express the following in terms of x : 10p

Exercise 10.1 | Q 7.2

If log 10 x = p, express the following in terms of x: 10p+1

Exercise 10.1 | Q 7.3

If log 10 x = p, express the following in terms of x: 102p-3

Exercise 10.1 | Q 7.4

If log 10 x = p, express the following in terms of x: 102-p

Exercise 10.1 | Q 8

If log 10 x = a, log 10 y = b and log 10 z = 2a - 3b, express z in terms of x and y.

Exercise 10.1 | Q 9.1

If Iog 10 a = x, log 10 b = y and log 10 c = z, find
102x-3 in term of a

Exercise 10.1 | Q 9.2

If Iog 10 a = x, log 10 b = y and log 10 c = z, find
103y -1 in terms of b

Exercise 10.1 | Q 9.3

If log 10 a = x,  and log 10 c = z, find
10^(x-y+z) in terms of a, b and c

Exercise 10.1 | Q 10.1

State true or false in the following:
If log 10 100 = 2, then 102 = 100

• True

• False

Exercise 10.1 | Q 10.2

State true or false in the following:
If log 10 p = q, then 10p = q

• True

• False

Exercise 10.1 | Q 10.3

State true or false in the following:
If 43 = 64, then log 3 64 = 4

• True

• False

Exercise 10.1 | Q 10.4

State true or false in the following:
If xy = z, then y = logxz

• True

• False

Exercise 10.1 | Q 10.5

State true or false in the following:
If log 2 8 = 3, then log 8 2 = (1)/(3)

• True

• False

Exercise 10.2

### Frank solutions for Class 9 Maths ICSE Chapter 10 Logarithms Exercise 10.2

Exercise 10.2 | Q 1.1

Express the following in terms of log 2 and log 3: log 36

Exercise 10.2 | Q 1.2

Express the following in terms of log 2 and log 3: log 54

Exercise 10.2 | Q 1.3

Express the following in terms of log 2 and log 3: log 144

Exercise 10.2 | Q 1.4

Express the following in terms of log 2 and log 3: log 216

Exercise 10.2 | Q 1.5

Express the following in terms of log 2 and log 3: log 648

Exercise 10.2 | Q 1.6

Express the following in terms of log 2 and log 3: log128

Exercise 10.2 | Q 2.1

Express the following in terms of log 5 and/or log 2: log20

Exercise 10.2 | Q 2.2

Express the following in terms of log 5 and/or log 2: log80

Exercise 10.2 | Q 2.3

Express the following in terms of log 5 and/or log 2: log125

Exercise 10.2 | Q 2.4

Express the following in terms of log 5 and/or log 2: log160

Exercise 10.2 | Q 2.5

Express the following in terms of log 5 and/or log 2: log500

Exercise 10.2 | Q 2.6

Express the following in terms of log 5 and/or log 2: log250

Exercise 10.2 | Q 3.1

Express the following in terms of log 2 and log 3: "log" root(3)(144)

Exercise 10.2 | Q 3.2

Express the following in terms of log 2 and log 3: "log"root(5)(216)

Exercise 10.2 | Q 3.3

Express the following in terms of log 2 and log 3: "log" root(4)(648)

Exercise 10.2 | Q 3.4

Express the following in terms of log 2 and log 3: "log"(26)/(51) - "log"(91)/(119)

Exercise 10.2 | Q 3.5

Express the following in terms of log 2 and log 3: "log"(225)/(16) - 2"log"(5)/(9) + "log"(2/3)^5

Exercise 10.2 | Q 4.1

Write the logarithmic equation for:

F = "G"("m"_1"m"_2)/"d"^2

Exercise 10.2 | Q 4.2

Write the logarithmic equation for:

E = (1)/(2)"m v"^2

Exercise 10.2 | Q 4.3

Write the logarithmic equation for:

n = sqrt(("M"."g")/("m".l)

Exercise 10.2 | Q 4.4

Write the logarithmic equation for:

V = (4)/(3)pi"r"^3

Exercise 10.2 | Q 4.5

Write the logarithmic equation for:

V = (1)/("D"l) sqrt("T"/(pi"r")

Exercise 10.2 | Q 5.01

Express the following as a single logarithm:
log 18 + log 25 - log 30

Exercise 10.2 | Q 5.02

Express the following as a single logarithm:
log 144 - log 72 + log 150 - log 50

Exercise 10.2 | Q 5.03

Express the following as a single logarithm:

2  "log"  3 - (1)/(2) "log"  16 + "log"  12

Exercise 10.2 | Q 5.04

Express the following as a single logarithm:

2 + 1/2 "log"  9 - 2  "log"  5

Exercise 10.2 | Q 5.05

Express the following as a single logarithm:

2"log"(9)/(5) - 3"log"(3)/(5) + "log"(16)/(20)

Exercise 10.2 | Q 5.06

Express the following as a single logarithm:

2"log"(15)/(18) - "log"(25)/(162) + "log"(4)/(9)

Exercise 10.2 | Q 5.07

Express the following as a single logarithm:

2"log" (16)/(25) - 3 "log" (8)/(5) + "log" 90

Exercise 10.2 | Q 5.08

Express the following as a single logarithm:

(1)/(2)"log"25 - 2"log"3 + "log"36

Exercise 10.2 | Q 5.09

Express the following as a single logarithm:

"log"(81)/(8) - 2"log"(3)/(5) + 3"log"(2)/(5) + "log"(25)/(9)

Exercise 10.2 | Q 5.1

Express the following as a single logarithm:

3"log"(5)/(8) + 2"log"(8)/(15) - (1)/(2)"log"(25)/(81) + 3

Exercise 10.2 | Q 6.1

Simplify the following:

2 "log" 5 +"log" 8 - (1)/(2) "log" 4

Exercise 10.2 | Q 6.2

Simplify the following:

2"log" 7 + 3 "log" 5 - "log"(49)/(8)

Exercise 10.2 | Q 6.3

Simplify the following:

3"log" (32)/(27) + 5 "log"(125)/(24) - 3"log" (625)/(243) + "log" (2)/(75)

Exercise 10.2 | Q 6.4

Simplify the following:

12"log" (3)/(2) + 7 "log" (125)/(27) - 5 "log" (25)/(36) - 7 "log" 25 + "log" (16)/(3)

Exercise 10.2 | Q 7.1

Solve the following:
log (3 - x) - log (x - 3) = 1

Exercise 10.2 | Q 7.2

Solve the following:
log(x2 + 36) - 2log x = 1

Exercise 10.2 | Q 7.3

Solve the following:
log 7 + log (3x - 2) =  log (x + 3) + 1

Exercise 10.2 | Q 7.4

Solve the following:
log ( x + 1) + log ( x - 1) = log 11 + 2 log 3

Exercise 10.2 | Q 7.5

Solve the following:
log 4 x + log 4 (x-6) = 2

Exercise 10.2 | Q 7.6

Solve the following:
log 8 (x2 - 1) - log 8 (3x + 9) = 0

Exercise 10.2 | Q 7.7

Solve the following:
log (x + 1) + log (x - 1) = log 48

Exercise 10.2 | Q 7.8

Solve the following:
log_2x + log_4x + log_16x = (21)/(4)

Exercise 10.2 | Q 8.1

Solve for x: log (x + 5) = 1

Exercise 10.2 | Q 8.2

Solve for x: ("log"27)/("log"243) = x

Exercise 10.2 | Q 8.3

Solve for x: ("log"81)/("log"9) = x

Exercise 10.2 | Q 8.4

Solve for x: ("log"121)/("log"11) = logx

Exercise 10.2 | Q 8.5

Solve for x: ("log"125)/("log"5) = logx

Exercise 10.2 | Q 8.6

Solve for x: ("log"128)/("log"32) = x

Exercise 10.2 | Q 8.7

Solve for x: ("log"1331)/("log"11) = logx

Exercise 10.2 | Q 8.8

Solve for x: ("log"289)/("log"17) = logx

Exercise 10.2 | Q 9

Express log103 + 1 in terms of log10x.

Exercise 10.2 | Q 10.1

State, true of false:
log (x + y) = log xy

• True

• False

Exercise 10.2 | Q 10.2

State, true of false:
log 4 x log 1 = 0

• True

• False

Exercise 10.2 | Q 10.3

State, true of false:
logba =-logab

• True

• False

Exercise 10.2 | Q 10.4

State, true of false:

If ("log"49)/("log"7) = log y, then y = 100.

• True

• False

Exercise 10.2 | Q 11.1

If log 16 = a, log 9 = b and log 5 = c, evaluate the following in terms of a, b, c: log 12

Exercise 10.2 | Q 11.2

If log 16 = a, log 9 = b and log 5 = c, evaluate the following in terms of a, b, c: log 75

Exercise 10.2 | Q 11.3

If log 16 = a, log 9 = b and log 5 = c, evaluate the following in terms of a, b, c: log 720

Exercise 10.2 | Q 11.4

If log 16 = a, log 9 = b and log 5 = c, evaluate the following in terms of a, b, c: log 2.25

Exercise 10.2 | Q 11.5

If log 16 = a, log 9 = b and log 5 = c, evaluate the following in terms of a, b, c: "log"2(1)/(4)

Exercise 10.2 | Q 12

If log x = p + q and log y = p - q, find the value of log (10x)/y^2 in terms of p and q.

Exercise 10.2 | Q 13

If log a = p and log b = q, express "a"^3/"b"^2 in terms of p and q.

Exercise 10.2 | Q 14

If log x = A + B and log y = A-B, express the value of "log" x^2/(10y) in terms of A and B.

Exercise 10.2 | Q 15.1

If log x = a and log y = b, write down
10a-1 in terms of x

Exercise 10.2 | Q 15.2

If log x = a and log y = b, write down
102b in terms of y

Exercise 10.2 | Q 16.1

If log 3 m = x and log 3 n = y, write down
32x-3 in terms of m

Exercise 10.2 | Q 16.2

If log 3 m = x and log 3 n = y, write down
3^(1-2y+3x) in terms of m an n

Exercise 10.2 | Q 17.1

If 2 log x + 1 = 40, find: x

Exercise 10.2 | Q 17.2

If 2 log x + 1 = 40, find: log 5x

Exercise 10.2 | Q 18.1

If log1025 = x and log1027 = y; evaluate without using logarithmic tables, in terms of x and y: log105

Exercise 10.2 | Q 18.2

If log1025 = x and log1027 = y; evaluate without using logarithmic tables, in terms of x and y: log103

Exercise 10.2 | Q 19.1

If log 2 = 0.3010, log 3 = 0.4771 and log 5 = 0.6990, find the values of: log18

Exercise 10.2 | Q 19.2

If log 2 = 0.3010, log 3 = 0.4771 and log 5 = 0.6990, find the values of: log45

Exercise 10.2 | Q 19.3

If log 2 = 0.3010, log 3 = 0.4771 and log 5 = 0.6990, find the values of: log540

Exercise 10.2 | Q 19.4

If log 2 = 0.3010, log 3 = 0.4771 and log 5 = 0.6990, find the values of: "log" sqrt(72)

Exercise 10.2 | Q 20

If 2 log y - log x - 3 = 0, express x in terms of y.

Exercise 10.2 | Q 21.1

If log 2 = x and log 3 = y, find the value of each of the following on terms of x and y: log60

Exercise 10.2 | Q 21.2

If log 2 = x and log 3 = y, find the value of each of the following on terms of x and y: log1.2

Exercise 10.2 | Q 22.1

If log 4 = 0.6020, find the value of each of the following: log8

Exercise 10.2 | Q 22.2

If log 4 = 0.6020, find the value of each of the following: log2.5

Exercise 10.2 | Q 23.1

If log 8 = 0.90, find the value of each of the following: log4

Exercise 10.2 | Q 23.2

If log 8 = 0.90, find the value of each of the following: "log"sqrt(32)

Exercise 10.2 | Q 24.1

If log 27 = 1.431, find the value of the following: log 9

Exercise 10.2 | Q 24.2

If log 27 = 1.431, find the value of the following: log300

Exercise 10.2 | Q 25

If x2 + y2 = 6xy, prove that "log"((x - y)/2) = (1)/(2) (log x + log y)

Exercise 10.2 | Q 26

If x2 + y2 = 7xy, prove that "log"((x - y)/3) = (1)/(2) (log x + log y)

Exercise 10.2 | Q 27

Find x and y, if ("log"x)/("log"5) = ("log"36)/("log"6) = ("log"64)/("log"y)

Exercise 10.2 | Q 28

If "log" x^2 - "log"sqrt(y) = 1, express y in terms of x. Hence find y when x = 2.

Exercise 10.2 | Q 29.1

If 2 log x + 1 = log 360, find: x

Exercise 10.2 | Q 29.2

If 2 log x + 1 = log 360, find: log(2 x -2)

Exercise 10.2 | Q 29.3

If 2 log x + 1 = log 360, find: log (3 x2 - 8)

Exercise 10.2 | Q 30

If x + log 4 + 2 log 5 + 3 log 3 + 2 log 2 = log 108, find the value of x.

Exercise 10.2 | Q 31.1

Simplify: log a2 + log a-1

Exercise 10.2 | Q 31.2

Simplify: log b ÷ log b2

Exercise 10.2 | Q 32.1

Find the value of:

("log"sqrt(8))/(8)

Exercise 10.2 | Q 32.2

Find the value of:

("log"sqrt(27) + "log"8 + "log"sqrt(1000))/("log"120)

Exercise 10.2 | Q 32.3

Find the value of:

("log"sqrt125 - "log"sqrt(27) - "log"sqrt(8))/("log"6 - "log"5)

Exercise 10.2 | Q 33

If a = "log" 3/5, "b" = "log" 5/4 and "c" = 2 "log" sqrt(3/4, prove that 5a+b-c = 1

Exercise 10.2 | Q 34.1

Express the following in a form free from logarithm:
3 log x - 2 log y = 2

Exercise 10.2 | Q 34.2

Express the following in a form free from logarithm:
2 log x + 3 log y = log a

Exercise 10.2 | Q 34.3

Express the following in a form free from logarithm:
m log x - n log y = 2 log 5

Exercise 10.2 | Q 34.4

Express the following in a form free from logarithm:
2"log" x + 1/2"log" y = 1

Exercise 10.2 | Q 34.5

Express the following in a form free from logarithm:
5 log m - 1 = 3 log n

Exercise 10.2 | Q 35

Prove that log (1 + 2 + 3) = log 1 + log 2 + log 3. Is it true for any three numbers x, y, z?

Exercise 10.2 | Q 36

Prove that (log a)2 - (log b)2 = "log"("a"/"b")."log"("ab")

Exercise 10.2 | Q 37

If a   b + b log  a - 1 = 0, then prove that ba.ab = 10

Exercise 10.2 | Q 38

If log (a + 1) = log (4a - 3) - log 3; find a.

Exercise 10.2 | Q 39

Prove that log 10 125 = 3 (1 - log 10 2)

Exercise 10.2 | Q 40

Prove that ("log"_"p" x)/("log"_"pq" x) = 1 + logp q

Exercise 10.2 | Q 41.1

Prove that: (1)/("log"_2 30) + (1)/("log"_3 30) + (1)/("log"_5 30) = 1

Exercise 10.2 | Q 41.2

Prove that: (1)/("log"_8 36) + (1)/("log"_9 36) + (1)/("log"_18 36) = 2

Exercise 10.2 | Q 42

If "a" = "log""p"^2/"qr", "b" = "log""q"^2/"rp", "c" = "log""r"^2/"pq", find the value of a + b + c.

Exercise 10.2 | Q 43

If a = log 20 b = log 25 and 2 log (p - 4) = 2a - b, find the value of 'p'.

## Chapter 10: Logarithms

Exercise 10.1Exercise 10.2

## Frank solutions for Class 9 Maths ICSE chapter 10 - Logarithms

Frank solutions for Class 9 Maths ICSE chapter 10 (Logarithms) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CISCE Class 9 Maths ICSE solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 9 Maths ICSE chapter 10 Logarithms are Introduction of Logarithms, Interchanging Logarithmic and Exponential Forms, Laws of Logarithm, Expansion of Expressions with the Help of Laws of Logarithm, More About Logarithm, Logarithmic to Exponential, Exponential to Logarithmic, Quotient Law, Power Law, Product Law.

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