Logarithms — Multiple Choice Questions

The relation $\sqrt[3]{64} = 4$ in logarithmic form is:

1. log₆₄ 4 = 3

2. $\log_4 \space 64 = \dfrac{1}{3}$

3. $\log_{64} \space 4 = \dfrac{1}{3}$

4. $\log_{64} \space \dfrac{1}{3} = 4$

Answer

Given,

$\Rightarrow \sqrt[3]{64} = 4 \\[1em] \Rightarrow (64)^{\dfrac{1}{3}} = 4 \\[1em] \Rightarrow \log_{64} \space 4 = \dfrac{1}{3}.$

Hence, option 3 is the correct option.

The relation log₃ 243 = 5 in exponential form is:

1. 5³ = 243

2. 3⁵ = 243

3. $243^{\dfrac{1}{3}} = 5$

4. 243³ = 5

Answer

Given,

⇒ log₃ 243 = 5

⇒ 243 = 3⁵

⇒ 3⁵ = 243.

Hence, option 2 is the correct option.

$\log_{\sqrt{2}} \space \Big(4\sqrt{2}\Big) =$

1. 8

2. 6

3. 4

4. 5

Answer

Given,

$\Rightarrow \log_{\sqrt{2}} \space \Big(4\sqrt{2}\Big)$

Let,

$\Rightarrow \log_{\sqrt{2}} \Big(4\sqrt{2}\Big) = y \\[1em] \Rightarrow (\sqrt{2})^{y} = 4\sqrt{2} \\[1em] \Rightarrow [(2)^{\dfrac{1}{2}}]^{y} = 2^2 \times 2^{\dfrac{1}{2}} \\[1em] \Rightarrow (2)^{\dfrac{y}{2}} = 2^{2 + \dfrac{1}{2}} \\[1em] \Rightarrow (2)^{\dfrac{y}{2}} = 2^{\dfrac{4 + 1}{2}} \\[1em] \Rightarrow (2)^{\dfrac{y}{2}} = 2^{\dfrac{5}{2}} \\[1em] \Rightarrow \dfrac{y}{2} = \dfrac{5}{2} \\[1em] \Rightarrow y = \dfrac{5}{2} \times 2 \\[1em] \Rightarrow y = 5.$

Hence, option 4 is the correct option.

$\log_3 \Big(\dfrac{1}{27}\Big) =$

1. 3

2. $\dfrac{1}{3}$

3. $-\dfrac{1}{3}$

4. -3

Answer

Given,

$\Rightarrow \log_3 \space \Big(\dfrac{1}{27}\Big)$

Let,

$\Rightarrow \log_3 \space \Big(\dfrac{1}{27}\Big) = x \\[1em] \Rightarrow \Big(\dfrac{1}{27}\Big) = 3^x \\[1em] \Rightarrow 3^{-3} = 3^x \\[1em] \Rightarrow x = -3.$

Hence, option 4 is the correct option.

log 5 + 2log 3 =

1. log 11

2. log 45

3. log 30

4. log 14

Answer

Given,

⇒ log 5 + 2log 3

⇒ log 5 + log 3²

⇒ log 5 + log 9

⇒ log (5 × 9)

⇒ log 45.

Hence, option 2 is the correct option.

log(1 × 2 × 3) =

1. log 5

2. log 1 × log 2 × log 3

3. log 1 + log 2 + log 3

4. log 9

Answer

Given,

⇒ log (1 × 2 × 3)

⇒ log 1 + log 2 + log 3.

Hence, option 3 is the correct option.

The value of log 0.0001 to the base 0.1 is:

1. 4

2. 3

3. $\dfrac{1}{4}$

4. $\dfrac{1}{3}$

Answer

Let,

⇒ log_0.1 (0.0001) = x

⇒ 0.0001 = 0.1^x

⇒ $\dfrac{1}{10000} = \Big(\dfrac{1}{10}\Big)^x$

⇒ $\dfrac{1}{10^4} = \Big(\dfrac{1}{10}\Big)^x$

⇒ $\Big(\dfrac{1}{10}\Big)^4 = \Big(\dfrac{1}{10}\Big)^x$

Equating the exponents,

⇒ x = 4.

Hence, option 1 is the correct option.

If log_x 243 = 5, then x =

1. 5

2. 3

3. $\dfrac{1}{3}$

4. 1

Answer

Given,

⇒ log_x 243 = 5

⇒ 243 = x⁵

⇒ 3⁵ = x⁵

⇒ x = 3.

Hence, option 2 is the correct option.

If log₅ (8x - 3) = 3, then x =

1. 8

2. 16

3. 32

4. 40

Answer

Given,

⇒ log₅ (8x - 3) = 3

⇒ (8x - 3) = 5³

⇒ 8x - 3 = 125

⇒ 8x = 125 + 3

⇒ 8x = 128

⇒ x = $\dfrac{128}{8}$

⇒ x = 16.

Hence, option 2 is the correct option.

log₉ 27 =

1. 3

2. $\dfrac{1}{3}$

3. $\dfrac{2}{3}$

4. $\dfrac{3}{2}$

Answer

Let,

⇒ log₉ 27 = x

⇒ 27 = 9^x

⇒ 3³ = (3²)^x

⇒ 3³ = 3^2x

Equating the exponents,

⇒ 2x = 3

⇒ x = $\dfrac{3}{2}$.

Hence, option 4 is the correct option.

$\dfrac{\log \space 27}{\log \space 9} =$

1. $\dfrac{3}{2}$

2. $\dfrac{2}{3}$

3. 3

4. 2

Answer

Given,

$\Rightarrow \dfrac{\log \space 27}{\log \space 9} \\[1em] \Rightarrow \dfrac{\log \space 3^3}{\log \space 3^2} \\[1em] \Rightarrow \dfrac{3\log \space 3}{2\log \space 3} \\[1em] \Rightarrow \dfrac{3}{2}.$

Hence, option 1 is the correct option.

If log_x 0.0016 = 4, then the value of x is:

1. 2

2. 0.2

3. 0.1

4. 4

Answer

Given,

⇒ log_x 0.0016 = 4

⇒ 0.0016 = x⁴

⇒ (0.2)⁴ = x⁴

⇒ x = 0.2

Hence, option 2 is the correct option.

If log₁₀ 2 = 0.3, then log₁₀ 8 =

1. 0.9

2. 0.6

3. 1.2

4. none of these

Answer

Given,

⇒ log₁₀ 8

⇒ log₁₀ 2³

⇒ 3log₁₀ 2

⇒ 3 × 0.3

⇒ 0.9

Hence, option 1 is the correct option.

$\log_2 \space \log_{\sqrt{2}} \log_3 81 =$

1. 1

2. 2

3. $\dfrac{1}{\sqrt{2}}$

4. $\dfrac{1}{2}$

Answer

Given,

$\Rightarrow \log_2 \space \log_{\sqrt{2}} \space \log_3 81 \\[1em] \Rightarrow \log_2 \space \log_{\sqrt{2}} \space \log_3 3^4 \\[1em] \Rightarrow \log_2 \space \log_{\sqrt{2}} \space 4\log_3 3 \\[1em] \Rightarrow \log_2 \space (\log_{\sqrt{2}} \space 4) \\[1em] \Rightarrow \log_2 \space \Big(\dfrac{\log \space 4}{\log \space \sqrt{2}}\Big) \\[1em] \Rightarrow \log_2 \space \Big(\dfrac{\log \space 2^2}{\log \space \sqrt{2}}\Big) \\[1em] \Rightarrow \log_2 \space \Big(\dfrac{2\log \space 2}{\dfrac{1}{2}\log \space 2}\Big) \\[1em] \Rightarrow \log_2 \space {\Big(\dfrac{2}{\dfrac{1}{2}}\Big)} \\[1em] \Rightarrow \log_2 \space 4 \\[1em] \Rightarrow \log_2 \space 2^2 \\[1em] \Rightarrow 2\log_2 \space 2 \\[1em] \Rightarrow 2.$

Hence, option 2 is the correct option.

If log₁₀ 2 = 0.3010 and log₁₀ 3 = 0.4771, then the value of log₁₀ 72 =

1. 1.8572

2. 0.8572

3. 0.5872

4. 1.5872

Answer

Given,

⇒ log₁₀ 72

⇒ log₁₀ (9 × 8)

⇒ log₁₀ 9 + log₁₀ 8

⇒ log₁₀ 3² + log₁₀ 2 ³

⇒ 2log₁₀ 3 + 3log₁₀ 2

⇒ 2(0.4771) + 3(0.3010)

⇒ 0.9542 + 0.9030

⇒ 1.8572

Hence, option 1 is the correct option.