Discrete Mathematics Questions and Answers – Discrete Probability – Logarithmic Series

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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 ax ≡ 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.
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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 alogax = 348?
a) 7.1
b) 4.5
c) 6.2
d) 4.8
View Answer

Answer: d
Explanation: Since, alogax = 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 logax = logay 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 2x + 3 = 5x + 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 2x + 3 = 5x + 2. By taking ln of both sides: ln (2x + 3) = ln (5x + 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 log2/3z, 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: 16x – 4x – 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: 16x – 4x – 9 = 0. Since 16x = (4x)2, the equation may be written as: (4x)2 – 4x – 9 = 0. Let t = 3x 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 4x = ln [ (1 + \(\sqrt{37}\)) / 2] ⇒ x = ln [ ( 1 + \(\sqrt{37}\))/2] / ln 3.

9. Transform 54y = 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: ay = x ⇔ y = loga (x) to write 3x = m as a logarithm: y = log54 (n+1).
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10. If loga\((\frac{1}{8}) = -\frac{3}{4}\), 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 = 329/5 = 29 = 512.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn