This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Logarithmic Series”.

1. Computation of the discrete logarithm is the basis of the cryptographic system _______

a) Symmetric cryptography

b) Asymmetric cryptography

c) Diffie-Hellman key exchange

d) Secret key cryptography

View Answer

Explanation: A discrete logarithm modulo of an integer to the base is an integer such that a

^{x}≡ b (mod g). The problem of computing the discrete logarithm is a well-known challenge in the field of cryptography and is the basis of the cryptographic system i.e., the Diffie-Hellman key exchange.

2. Solve the logarithmic function of ln(\(\frac{1+5x}{1+3x}\)).

a) 2x – 8x^{2} + \(\frac{152x^3}{3}\) – …

b) x^{2} + \(\frac{7x^2}{2} – \frac{12x^3}{5}\) + …

c) x – \(\frac{15x^2}{2} + \frac{163x^3}{4}\) – …

d) 1 – \(\frac{x^2}{2} + \frac{x^4}{4}\) – …

View Answer

Explanation: To solve the logarithmic function ln(\(\frac{1+5x}{1+3x}\)) = ln(1+5x) – ln(1+3x) = (5x – \(\frac{(5x)^2}{2} + \frac{(5x)^3}{3}\) – …) – (3x – \(\frac{(3x)^2}{2} + \frac{(3x)^3}{3}\) – …) = 2x – 8x

^{2}+ \(\frac{152x^3}{3}\) – …

3. Determine the logarithmic function of ln(1+5x)^{-5}

a) 5x + \(\frac{25x^2}{2} + \frac{125x^3}{3} + \frac{625x^4}{4}\) …

b) x – \(\frac{25x^2}{2} + \frac{625x^3}{3} – \frac{3125x^4}{4}\) …

c) \(\frac{125x^2}{3} – 625x^3 + \frac{3125x^4}{5}\) …

d) -25x + \(\frac{125x^2}{2} – \frac{625x^3}{3} + \frac{3125x^4}{4}\) …

View Answer

Explanation: Apply the logarithmic law, that is logax = xlog(a). Now the function is ln(1+5x)

^{-5}= -5log(1+5x). By taking the series = -5(5x – \(\frac{(5x)^2}{2} + \frac{(5x)^3}{3} – \frac{(5x)^4}{4}\) + …) = -25x + \(\frac{125x^2}{2} – \frac{625x^3}{3} + \frac{3125x^4}{4}\) …

4. Find the value of x: 3 x^{2} a^{logax} = 348?

a) 7.1

b) 4.5

c) 6.2

d) 4.8

View Answer

Explanation: Since, a

^{logax}= x . The given equation may be written as: 3x

^{2}x = 348 ⇒ x = (116)

^{1/3}= 4.8.

5. Solve for x: log_{2}(x^{2}-3x)=log_{2}(5x-15).

a) 2, 5

b) 7

c) 23

d) 3, 5

View Answer

Explanation: By using the property if log

_{a}x = log

_{a}y then x=y, gives 2x

^{2}-3x=10-6x. Now, to solve the equation x

^{2}-3x-5x+15=0 ⇒ x

^{2}-8x+15 ⇒ x=3, x=5

For x=3: log

_{2}(3

^{2}-3*3) = log

_{2}(5*3-15) ⇒ true

For x=5: log

_{2}(5

^{2}-3*5) = log

_{2}(5*5-15) ⇒ true

The solutions to the equation are : x=3 and x=5.

6. Solve for x the equation 2^{x + 3} = 5^{x + 2}.

a) ln (24/8)

b) ln (25/8) / ln (2/5)

c) ln (32/5) / ln (2/3)

d) ln (3/25)

View Answer

Explanation: Given that 2

^{x + 3}= 5

^{x + 2}. By taking ln of both sides: ln (2

^{x + 3}) = ln (5

^{x + 2})

⇒ (x + 3) ln 2 = (x + 2) ln 5

⇒ x ln 2 + 3 ln 2 = x ln 5 + 2 ln 5

⇒ x ln 2 – x ln 5 = 2 ln 5 – 3 ln 2

⇒ x = ( 2 ln 5 + 3 ln 2 ) / (ln 2 – ln 5) = ln (5

^{2}/ 2

^{3}) / ln (2/5) = ln (25/8) / ln (2/5).

7. Given: log_{4} z = B log_{2/3}z, for all z > 0. Find the value of constant B.

a) 2/(3!*ln(2))

b) 1/ln(7)

c) (4*ln(9))

d) 1/(2*ln(3))

View Answer

Explanation: By using change of base formula we can have ln (x) / ln(4) = B ln(x) / ln(2/3) ⇒

B = 1/(2*ln(3)).

8. Evaluate: 16^{x} – 4^{x} – 9 = 0.

a) ln [( 5 + \(\sqrt{21}\)) / 2] / ln 8

b) ln [( 2 + \(\sqrt{33}\)) / 2] / ln 5

c) ln [( 1 + \(\sqrt{37}\)) / 2] / ln 4

d) ln [( 1 – \(\sqrt{37}\)) / 2] / ln 3

View Answer

Explanation: Given: 16

^{x}– 4

^{x}– 9 = 0. Since 16

^{x}= (4

^{x})

^{2}, the equation may be written as: (4

^{x})

^{2}– 4

^{x}– 9 = 0. Let t = 3

^{x}and so t: t

^{2}– t – 9 = 0 which gives t: t = (1 + \(\sqrt{37}\)) / 2 and (1 – \(\sqrt{37}\)) / 2

Since t = 4x, the acceptable solution is y = (1 + \(\sqrt{37}\)) / 2 ⇒ 4x = (1 + \(\sqrt{37}\))/2. By using ln on both sides: ln 4

^{x}= ln [ (1 + \(\sqrt{37}\)) / 2] ⇒ x = ln [ ( 1 + \(\sqrt{37}\))/2] / ln 3.

9. Transform 54^{y} = n+1 into equivalent a logarithmic expression.

a) log_{12} (n+1)

b) log_{41} (n^{2})

c) log_{63} (n)

d) log_{54} (n+1)

View Answer

Explanation: By using the equivalent expression: a

^{y}= x ⇔ y = log

_{a}(x) to write 3

^{x}= m as a logarithm: y = log

_{54}(n+1).

10. If log_{a}\((\frac{1}{8}) = -\frac{3}{4}\), than what is x?

a) 287

b) 469

c) 512

d) 623

View Answer

Explanation: By using exponential form: a

^{-5/9}= 2/8. Now, raise both sides of the above equation to the power -9/5: (x

^{-5/9})

^{-9/5}= (1/32)

^{-9/5}. By simplifying we get, a = 32

^{9/5}= 2

^{9}= 512.

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