Special Functions MCQ Quiz - Objective Question with Answer for Special Functions - Download Free PDF

Concept:

• log A + log B = log AB
• log A - log B = log(A/B)
• log a^k = k log a

Calculation:

Given,

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

∴ The correct answer is option (1).

Explanation:

Since, {x}+[x] = x

⇒x – [x] = {x}

⇒ 0≤ x – [x]

⇒ –1 ≤ [x] –x ≤ 0

But x is positive and non-integer; then

⇒ –1

⇒ –1

⇒ [y] = –1 

∴ Option (a) is correct.

Given:

8k = 2

Formula used:

log_b a = c ^{if and only if} b^c = a

Calculation:

Given: 8k = 2

⇒ log₈ 2 = x

∴ The correct answer is option (3).

Given:

log₁₀ 10000 = ?

Formula used:

log_b a = c ⟺ b^c = a

Calculation:

log₁₀ 10000 = ?

10000 = 10⁴

⇒ log₁₀ 10000 = 4

∴ The correct answer is option (4).

Explanation -

We have,

⇒ 2 × 2^2x-1 + 2^2x-1 =  × 3 + 

⇒ 2^2x-1 (2 + 1) =  (3 + 1)

⇒ 22x-1 × 3 =  × 4

⇒ 22x-3 = 

⇒  =  

⇒  =  

⇒  = 0 

⇒ 

​Hence Option (2) is correct.

Concept:

Logarithm properties:  

Product rule: The log of a product equals the sum of two logs.

Quotient rule: The log of a quotient equals the difference of two logs.

Power rule: In the log of power the exponent becomes a coefficient.

Formula of Logarithms:

If  then x = ab (Here a ≠ 1 and a > 0)

Calculation:

Given: 

        (∵ )

               (∵ ) 

Concept:

, where  and a > 0 and x be any number.

Calculation:

Given: 92^1/5 = 4.

As we know that, 

Comparing 92^1/5 = 4 with  we have,

Here, a = 92, b = 1 / 5 and x = 4.

So, the logarithmic form of 92^1/5 = 4 is .

Concept:

Logarithm properties

1. Product rule: The log of a product equals the sum of two logs.

2. Quotient rule: The log of a quotient equals the difference of two logs.

3. Power rule: In the log of a power the exponent becomes a coefficient.

4. Change of base rule

If m = n;
⇒

Calculation:

Here, we have to find the value of

= log₇ log₇ (7^1/2 × 7^1/4 × 7^1/8)

= log₇ log₇ (7(^{1/2 + 1/4 + 1/8)})

= log₇ log₇ (7^{(4 + 2 + 1)/8})

= log₇ log₇ (7^7/8)

From power rule;

= log₇ (7/8) log₇7

= log₇ (7/8) × 1 = log₇ (7/8) = log₇ 7 – log₇ 8

= 1 – log₇ 8 = 1 – log₇ 2³

= 1 – 3 log₇ 2

Concept:

Logarithm properties:  

Product rule: The log of a product equals the sum of two logs.

Quotient rule: The log of a quotient equals the difference of two logs.

Power rule: In the log of power the exponent becomes a coefficient.

Formula of Logarithms:

If  then x = ab (Here a ≠ 1 and a > 0)

Calculation:

Given: \(\rm \log_{4}{(x^{2} - 1)} - \log_{4} (x + 1) = 1\)

        (∵ )

⇒ (x - 1) = 4

∴ x = 5

Concept:

Logarithms:

• If ab = x, then we say that loga x = b.
• log_a a = 1.
• log_a (xy) = log_a x + log_a y.

Calculation:

We know that 80 = 23 × 10.

Changing the given logarithms to log 2, we get:

log₁₀ 80

= log₁₀ (23 × 10)

= log10 2³ + log10 10

= 3 (log10 2) + 1

= 3(0.3010) + 1

= 1.9030.

Given:

5x-1 = (2.5)log105

Formula Used:

If a^x = n then x = log_aⁿ

log_a^b = log_e^b/log_e^a

Calculation:

We have 5x-1 = (2.5)log105

⇒ (2.5)log105 = 5x-1 

⇒ log₁₀⁵  = log_2.5^5x-1 

⇒ log105  = (x - 1) log2.55

⇒ (x - 1) = (log105)/(log2.55)

⇒ (x - 1) = log₁₀^2.5

⇒ x = log102.5 _{+ 1}

⇒ x = log102.5 ₊ log₁₀¹⁰

⇒ x = log1010 × 2.5

⇒ x = log1025

⇒ x = log105²

⇒ x = 2log10​5

∴ The value of x is 2log10​5.

Concept:

Logarithm properties

1. Product rule: The log of a product equals the sum of two logs.

2. Quotient rule: The log of a quotient equals the difference of two logs.

3. Power rule: In the log of a power the exponent becomes a coefficient.

4. Change of base rule

If m = n;

⇒ 

5. 

Calculation:

Here, we have to find the value of 

Now,

 = log₃ log₃ (3^1/2 × 3^1/4)

= log₃ log₃ (3^{(1/2 + 1/4)})

= log₃ log₃ (3^{(2 + 1)/4})

= log₃ log₃ (3^3/4)

From power rule;

= log₃ [(3/4)× log₃3]                [∵ log_a (m) ⁿ = n × log_a (m)]

= log₃ (3/4)                  (∵ log_m m = 1)

= log₃ (3/4) = log₃ 3 – log₃ 4

= 1 – log₃ 4 = 1 – log₃ 2²

= 1 – 2 log₃ 2

Concept:

Formula used:

• Log_a M + log_a N = log_a (MN)

Factorial:

• n! = 1 × 2 × 3 × ⋯ × (n – 1) × n

Calculation:

Using 

= log_N (2 × 3 × ⋯ × 100)

= log_N (100!)

Concept:

Logarithm Rule

log mⁿ = n log m

Calculations:

Let x, y, z are three consecutive positive integers.

⇒ y = x + 1 and z = y + 1

⇒ z = x + 2

Consider, log (1 + xz)

= log [1 + x(x+2)]

= log [1 + x² + 2x]

= log (1 + x)²

= 2 log (1 + x)

= 2 log y

Hence, If x, y, z are three consecutive positive integers, then log (1 + xz) is 2 log y

CONCEPT:

• By base changing theorem we know that .
• log _x (x) = 1

CALCULATION:

Given that 

By base changing we can write it as - 

⇒ log₆₀9 + log6016 +log6025

By product rule of logarithms we can write it again as - 

⇒ log₆₀(9 x 16 x 25) = log₆₀(3600)

⇒ log₆₀ (60)² = 2 log₆₀(60) = 2

Therefore, option (3) is the correct answer.