Logarithmic Series - Discrete Mathematics Questions and Answers

This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Discrete Probability – 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

Answer: c
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 – 8x² + \(\frac{152x^3}{3}\) – …
b) x² + \(\frac{7x^2}{2} – \frac{12x^3}{5}\) + …
c) x – \(\frac{15x^2}{2} + \frac{163x^3}{4}\) – …
d) 1 – \(\frac{x^2}{2} + \frac{x^4}{4}\) – …
View Answer

Answer: a
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 – 8x² + \(\frac{152x^3}{3}\) – …

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

Answer: d
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 x² a^log_ax = 348?
a) 7.1
b) 4.5
c) 6.2
d) 4.8
View Answer

Answer: d
Explanation: Since, a^log_ax = x. The given equation may be written as: 3x² x = 348 ⇒ x = (116)^1/3 = 4.8.

5. Solve for x: log₂(x²-3x)=log₂(5x-15).
a) 2, 5
b) 7
c) 23
d) 3, 5
View Answer

Answer: d
Explanation: By using the property if log_ax = log_ay then x=y, gives 2x²-3x=10-6x. Now, to solve the equation x²-3x-5x+15=0 ⇒ x²-8x+15 ⇒ x=3, x=5
For x=3: log₂(3²-3*3) = log₂(5*3-15) ⇒ true
For x=5: log₂(5²-3*5) = log₂(5*5-15) ⇒ true
The solutions to the equation are : x=3 and x=5.

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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

Answer: b
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 (5² / 2³) / ln (2/5) = ln (25/8) / ln (2/5).

7. Given: log₄ z = B log_2/3z, 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

Answer: d
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

Answer: c
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: t² – 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) log₁₂ (n+1)
b) log₄₁ (n²)
c) log₆₃ (n)
d) log₅₄ (n+1)
View Answer

Answer: d
Explanation: By using the equivalent expression: a^y = x ⇔ y = log_a (x) to write 3^x = m as a logarithm: y = log₅₄ (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

Answer: c
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 = 2⁹ = 512.

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