Nootan solutions for Mathematics [English] Class 9 ICSE chapter 7 - Logarithms [Latest edition]

Express the following in logarithmic form:

8³ = 512

Express the following in logarithmic form:

5³ = 125

Express the following in logarithmic form:

3² = 9

Express the following in logarithmic form:

10^–2 = 0.01

Express the following in logarithmic form:

`3^-3 = 1/27`

Express the following in logarithmic form:

`2^-5 = 1/32`

Express the following in exponential form:

log₅ 125 = 3

Express the following in exponential form:

log₆ 36 = 2

Express the following in exponential form:

log₄ 256 = 4

Express the following in exponential form:

log₂ 0.125 = –3

Express the following in exponential form:

`log_4  1/32 = - 5/2`

Express the following in exponential form:

log₁₀ 0.0001 = –4

Evaluate the following:

`log_(6sqrt(2)) 72`

Evaluate the following:

`log_(sqrt(3)) 27`

Evaluate the following by converting into exponential form:

log₅ 625

Evaluate the following by converting into exponential form:

log₂ 8

Evaluate the following by converting into exponential form:

log_0.5 64

Evaluate the following by converting into exponential form:

log₅ 0.04

Evaluate the following by converting into exponential form:

`log_(sqrt(3)) 9`

Evaluate the following by converting into exponential form:

`log_3  1/3`

Evaluate the following by converting into exponential form:

log₉ 243

Evaluate the following by converting into exponential form:

log₃₂ 2

Solve for x:

log₂ x = 3

Solve for x:

`log_125 x = 1/3`

Solve for x:

`log_(sqrt(2)) x = 4`

Solve for x:

log₃ x = –2

Solve for x:

log_x 81 = 4

Solve for x:

log_x 32 = –5

Solve for x:

log_x 8 = 1

Solve for x:

`log_x  1/2 = -1`

Solve for x:

`log_x 16 = 1/2`

Solve for x:

log₃ (x² – 19) = 4

Solve for x:

log (2x + 4) = 1

Solve for x:

log x = –1

If log₁₀ x = y, express `10^(3y  -  1)` in terms of x.

If log₁₀ x = a, log₁₀ y = b, express `10^(a - 1)` in terms of x.

If log₁₀ x = a, log₁₀ y = b, express `10^(3b - 2)` in terms of y.

If log₁₀ x = a, log₁₀ y = b, If log₁₀ c = 2a – b, find c in terms of x and y.

If log₂ x = a, log₃ y = a, find `24^(2a + 1)` in terms of x and y.

If log₂ y = x, log₃ z = x, express 12^x in terms of y and z.

If log₂ x = a, log₅ y = a, find `20^(2a  -  1)` in terms of x and y.

If log x = a, log y = b, express x³y² in terms of a and b.

Evaluate the following:

`1/2 log 625 + log 8 - 1/4 log 16`

Evaluate the following: 

`log 100 + 2 log 0.01 - 1/2 log 5^-4 + 2/3 log 8`

Evaluate the following:

log 5 + 3 log 2 – 2 log 6 + 2 log 3

Evaluate the following:

2 log 2 + 4 log 3 + log 5 – log 7.5 – log 2.16

Evaluate the following:

log₁₆ 32 – log₉ 27

Evaluate the following:

log₆₂₅ 125 + log₆₄ 16

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

log 24

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

log 72

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

log 13.5

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

log 180

Prove the following:

`2 log  15/4 + log  81/5 - 3 log  9/4 + log 100 = 3 + log 2`

Prove the following:

`log  125/147 = 3 - 3 log 2 - 2 log 7 - log 3`

Prove the following:

`log  35/33 - log  135/99 + log  24/7 = 3 log 2 - log 3`

Prove the following:

2 log₁₀ 20 + log 5 – log 2 = 3

Express the following as a single logarithm:

2 log 5 + 3 log 2 – 1

Express the following as a single logarithm:

2 log 3 + log 7 + 3 log 5 – 2

If log 2 = x and log 3 = y, express the following in terms of x and y.

log 250

If log 2 = x and log 3 = y, express the following in terms of x and y.

log 14.4

Prove that : `4^(log 9) = 3^(log 16)`.

If log x = a + b and log y = a – b, express log (x²y) in terms of a and b.

If log x = 3a + 2b and log y = 2a – b, express log (x³y²) in terms of a and b.

If `x = log  2/3, y = log  3/7, z = log  7/2`, find the value of x + y + z.

If `x = log  2/3, y = log  3/7, z = log  7/2`, find the value of `2^(x  +  y  +  z)`.

If `x = log  3/7, y = log  10/7, z = log  10/3`, find the value of x – y + z.

If `x = log  3/7, y = log  10/7, z = log  10/3`, find the value of `5^(x  -  y  +  z)`.

Find the value of x in the following:

log₁₀ x = log₁₀ 2 + 2

Find the value of x in the following:

log x = –2 + 3 log 2 – 5 log 3 + 2 log 72 + log 3

Find the value of x in the following:

log (x + 2) + log (x – 2) = log 2 + log 3 + 1

Find the value of x in the following:

log (x + 4) + log (x – 4) = 4 log 2 + log 3

Find the value of x in the following:

log (3x + 2) + log (3x – 2) = 1 + log 2 + log 7

Find the value of x in the following:

`log_2 x + log_8 x + log_32 x = 23/15`

Find the value of x in the following:

`log_3 x + log_9 x + log_81 x = 7/4`

Prove that: (1 + log_a b).log_ab x = log_a x.

Prove that: `(log x)^2 - (log y)^2 = log (x/y) · log (xy)`.

If `log  (x + y)/2 = (log x + log y)/2`, prove that x = y.

If `2 log  (x - y)/2 = log x + log y`, prove that x² + y² – 6xy = 0.

If log (x + y) = log x + log y, prove that: `y = x/(x - 1)`.

If log (x + y) = log x – log y, prove that `x = y^2/(1 - y)`.

If x² + y² = 34 xy, prove that `log ((x + y)/6) = 1/2 (log x + log y)`.

Show that: log_b a · log_c d = log_c a · log_b d.

If `1/(log_a x) + 1/(log_c x) = 2/(log_b x)`, then prove that b² = ac.

Show that: `1/(log_36 12) + 1/(log_6 12) + 1/(log_8 12) = 3`

If x = log_a (bc), y = log_b (ca), z = log_c (ab) then prove that `1/(1 + x) + 1/(1 + y) + 1/(1 + z)` = 1

If log (2x + 5) = 2, then x is equal to ______.

`log_(sqrt(2)) sqrt(8)` is equal to ______.

The value of 4^{log 3} – 3^{log 4} is ______.

If x = log 2, y = log 3, then log 12 is equal to ______.

If log_x 9 – log_x 3 – log_x 27 = 2, then x is equal to ______.

If log₂ x = a and log₅ y = a, then `100^(3a - 1)` is equal to ______.

log₄ 32 – log₈ 32 is equal to ______.

3 – 5 log₁₀ 2 is equal to ______.

If log (x + y) = log x + log y, then y is equal to ______.

If `log((x + y)/5) = (logx + logy)/2`, then correct relation is ______.

(i) log (m · n) = log m + log n

(ii) 3 – 5 log₁₀ 2 = log₁₀ 125 – log₁₀ 4

(i) log 288 = 5 log 2 + 2 log 3

(ii) 3 log 2 + 2 log 3 – 1 = log 72

(i) log m – log n = log (mⁿ)

(ii) If log₁₀ (x + 5) + log₁₀ (x – 5) = log 600, then x = 5

(i) 5^{log 4} = 4^{log 5}

(ii) If `x = log  1/2, y = log  2/3`, z = log 3, then x + y + z = 0