Frank solutions for Mathematics [English] Class 9 ICSE chapter 10 - Logarithms [Latest edition]

Express the following in the logarithmic form:
3³ = 27

Express the following in the logarithmic form:
5⁴ = 625

Express the following in the logarithmic form:
9⁰ = 1

Express the following in the logarithmic form:

`(1)/(8)` = 2^-3 

Express the following in the logarithmic form:
11² = 121

Express the following in the logarithmic form:

3^-2 = `(1)/(9)`

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

Express the following in the logarithmic form:
7⁰ = 1

Express the following in the logarithmic form:

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

Express the following in the logarithmic form:

9^-4 = `(1)/(6561)`

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

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

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

Express the following in the exponential form:

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

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

Express the following in the exponential form:

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

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

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

Express the following in the exponential form:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find the value of: log ₁₀ 1000

Find the value of: log ₃ 81

Find the value of: log ₅ 3125

Find the value of: log ₂ 128

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

Find the value of: log ₁₀ 0.0001

Find the value of: log ₅ 125

Find the value of: log ₈ 2

Find the value of `log_(1/2)16`.

Find the value of: log _0.0110

Find the value of: log ₃ 81

Find the value of: log ₅ `(1)/(25)`

Find the value of: log ₂ 8

Find the value of: log _a a³ 

Find the value of: log _0.1 10

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

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

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

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

If log ₁₀ x = a, express the following in terms of x: 10^{2a -3} 

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

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

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

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

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

If log ₁₀ x = p, express the following in terms of x: 10^2p-3 

If log ₁₀ x = p, express the following in terms of x: 10^2-p 

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

If Iog ₁₀ a = x, log ₁₀ b = y and log ₁₀ c = z, find 
10^2x-3 in term of a

If Iog ₁₀ a = x, log ₁₀ b = y and log ₁₀ c = z, find 
10^{3y -1} in terms of b

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

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

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

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

State true or false in the following:
If x^y = z, then y = log_xz

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Write the logarithmic equation for:

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

Write the logarithmic equation for:

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

Write the logarithmic equation for: 

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

Write the logarithmic equation for:

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

Write the logarithmic equation for: 

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

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

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

Express the following as a single logarithm:

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

Express the following as a single logarithm:

`2 + 1/2 "log"  9 - 2  "log"  5`

Express the following as a single logarithm:

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

Express the following as a single logarithm:

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

Express the following as a single logarithm:

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

Express the following as a single logarithm:

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

Express the following as a single logarithm:

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

Express the following as a single logarithm:

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

Simplify the following: 

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

Simplify the following:

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

Simplify the following:

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

Simplify the following:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Express log₁₀3 + 1 in terms of log₁₀x.

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

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

State, true of false:
log_ba =-log_ab

State, true of false:

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

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

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

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

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

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

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

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

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

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

If log x = a and log y = b, write down
10^2b in terms of y

If log ₃ m = x and log ₃ n = y, write down
3^2x-3 in terms of m

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify: log a² + log a^-1

Simplify: log b ÷ log b²

Find the value of:

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

Find the value of:

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

Find the value of:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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