Logarithms Questions and Answers

Q: Interval used to indicate interval of all integers between a and b is called

An integer interval holds only the whole numbers lying between its endpoints, rather than every real number there:\[ \{\, x \in \mathbb{Z} : a \le x \le b \,\} \]So between 2 and 5 it contains 2, 3, 4 and 5, whereas the closed interval \( [2,5] \) contains every real number in that range. Open and closed describe whether the endpoints are included; this option describes which kind of number is included.

Q: Inverse operation of logarithm is

Logarithm and exponential are inverse operations:\[ \log_{a}(a^{x}) = x, \qquad a^{\log_{a} x} = x \]A logarithm asks "to what power?", and exponentiation supplies that power - each undoes the other. Taking antilogs is exactly this exponentiation.

Q: Interval which does not include its endpoints, and is indicated with parentheses is known as

An open interval excludes its endpoints and is written with parentheses:\[ (a,b) = \{\, x \mid a < x < b \,\} \]The closed interval \( [a,b] \) uses square brackets and includes both ends.

Q: Laplace transform of function f(t)=cos(wt) is

The standard transform pair is\[ \mathcal{L}\{\cos \omega t\} = \frac{s}{s^{2}+\omega^{2}} \]while its companion is\[ \mathcal{L}\{\sin \omega t\} = \frac{\omega}{s^{2}+\omega^{2}} \]The denominators are identical; only the numerator distinguishes them - \( s \) for cosine, \( \omega \) for sine.

Q: The interval \( [a, +\infty) = \{\, x \mid x \geq a \,\} \) is

Read the two features separately.The square bracket at \( a \) means the endpoint is included (\( x \geq a \)), so the interval is closed at that end. It extends to \( +\infty \) with no upper endpoint, so it is unbounded.Hence it is closed and unbounded. Infinity is never itself included, which is why that side is always written with a parenthesis.

Q: A formula or rule for computing derivative of composition of two or more functions is known as

The chain rule differentiates a composition of functions:\[ \frac{d}{dx}f\big(g(x)\big) = f'\big(g(x)\big) \cdot g'(x) \]The product rule handles a product of functions, \( (uv)' = u'v + uv' \), which is a different situation entirely.

Q: If a number has no zero immediately after decimal point, characteristic is

For a number less than 1, the characteristic is negative: it is one more than the number of zeros immediately after the decimal point, taken with a minus sign.With no such zeros, as in 0.5 or 0.842, the characteristic is \( -1 \), usually written \( \bar{1} \).\[ \log 0.842 = \bar{1}.9253 = -1 + 0.9253 \]The mantissa stays positive; only the characteristic carries the sign.

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101.

Interval used to indicate interval of all integers between a and b is called

A.infinite interval
B.closed interval
C.integer interval
D.open interval

Answer: C

An integer interval holds only the whole numbers lying between its endpoints, rather than every real number there:

\[ \{\, x \in \mathbb{Z} : a \le x \le b \,\} \]

So between 2 and 5 it contains 2, 3, 4 and 5, whereas the closed interval \( [2,5] \) contains every real number in that range. Open and closed describe whether the endpoints are included; this option describes which kind of number is included.

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102.

Inverse operation of logarithm is

A.expansion
B.differentiation
C.integration
D.exponential

Answer: D

Logarithm and exponential are inverse operations:

\[ \log_{a}(a^{x}) = x, \qquad a^{\log_{a} x} = x \]

A logarithm asks "to what power?", and exponentiation supplies that power - each undoes the other. Taking antilogs is exactly this exponentiation.

Discuss in Forum

103.

Interval which does not include its endpoints, and is indicated with parentheses is known as

A.infinite interval
B.closed interval
C.integer interval
D.open interval

Answer: D

An open interval excludes its endpoints and is written with parentheses:

\[ (a,b) = \{\, x \mid a < x < b \,\} \]

The closed interval \( [a,b] \) uses square brackets and includes both ends.

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104.

Laplace transform of function f(t)=cos(wt) is

A.s/(s+w)
B.s/(s-w)
C.s/(s²+w²)
D.w/(s²+w²)

Answer: C

The standard transform pair is

\[ \mathcal{L}\{\cos \omega t\} = \frac{s}{s^{2}+\omega^{2}} \]

while its companion is

\[ \mathcal{L}\{\sin \omega t\} = \frac{\omega}{s^{2}+\omega^{2}} \]

The denominators are identical; only the numerator distinguishes them - \( s \) for cosine, \( \omega \) for sine.

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105.

The interval \( [a, +\infty) = \{\, x \mid x \geq a \,\} \) is

A.open bounded
B.close bounded
C.open unbounded
D.close unbounded

Answer: D

Read the two features separately.

The square bracket at \( a \) means the endpoint is included (\( x \geq a \)), so the interval is closed at that end. It extends to \( +\infty \) with no upper endpoint, so it is unbounded.

Hence it is closed and unbounded. Infinity is never itself included, which is why that side is always written with a parenthesis.

Discuss in Forum

106.

A formula or rule for computing derivative of composition of two or more functions is known as

A.product rule
B.composition rule
C.chain rule
D.distribution rule

Answer: C

The chain rule differentiates a composition of functions:

\[ \frac{d}{dx}f\big(g(x)\big) = f'\big(g(x)\big) \cdot g'(x) \]

The product rule handles a product of functions, \( (uv)' = u'v + uv' \), which is a different situation entirely.

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107.

If a number has no zero immediately after decimal point, characteristic is

A.1
B.-1
C.3

Answer: B

For a number less than 1, the characteristic is negative: it is one more than the number of zeros immediately after the decimal point, taken with a minus sign.

With no such zeros, as in 0.5 or 0.842, the characteristic is \( -1 \), usually written \( \bar{1} \).

\[ \log 0.842 = \bar{1}.9253 = -1 + 0.9253 \]

The mantissa stays positive; only the characteristic carries the sign.

Discuss in Forum

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