Logarithms Questions and Answers

Q: Logarithms having base ‘10’ are called

Logarithms to base 10 are the common logarithms, also called Briggsian after Henry Briggs, who computed the first base-10 tables.They are the convenient base for hand arithmetic because our numerals are base 10: \( \log 1000 = 3 \) exactly, and multiplying a number by 10 simply adds 1 to its logarithm, which is what makes the characteristic and mantissa split work.

Q: A concise, precise and convenient method to write very small or large numbers is called

Scientific notation writes a number as a figure between 1 and 10 multiplied by a power of ten, \( a \times 10^{n} \) with \( 1 \le a < 10 \).\[ 0.000581 \;=\; 5.81 \times 10^{-4}, \qquad 1563.4 \;=\; 1.5634 \times 10^{3} \]It is exactly this form that gives a logarithm its two parts: the exponent becomes the characteristic and the leading figure supplies the mantissa.

Q: Number whose logarithm is given is called

The antilogarithm is the number whose logarithm is given, so it undoes the logarithm:\[ \log x = y \quad\Longleftrightarrow\quad x = \operatorname{antilog} y = 10^{\,y} \]For instance \( \log 100 = 2 \), so \( \operatorname{antilog} 2 = 100 \). Common and natural logarithms name the base; antilogarithm names the inverse operation, which is why the other options do not fit.

Q: Napier's logarithm is also called

Napier's logarithms are the natural logarithms, taken to base \( e \) and written \( \ln x \).Common logarithms, to base 10, came slightly later and are named after Briggs. The mantissa is only the fractional part of a logarithm, and the antilogarithm is the inverse operation, so neither names a system of logarithms at all.

Q: Characteristic of log 0.000581 is

The number is below 1, so the characteristic is negative and its size is one more than the count of zeros between the decimal point and the first significant figure:\[ \text{characteristic} \;=\; -\left(\text{zeros after the point} + 1\right) \]0.000581 carries three such zeros, so\[ \text{characteristic} \;=\; -(3+1) \;=\; -4 \]Confirm it in standard form: \( 0.000581 = 5.81 \times 10^{-4} \), hence \( \log 0.000581 = -4 + \log 5.81 \). The positive options are the trap for dropping the sign that every number below 1 must carry.

Q: Decadic logarithm is also known as

The decadic logarithm is the logarithm to base 10, and its usual English name is the common logarithm, written \( \log_{10} \) or simply \( \log \).The natural logarithm takes base \( e \approx 2.718 \) and is written \( \ln \). "Decadic" comes from the Greek for ten, which is the clue to the base meant.

Important Instructions

11.

Logarithms having base ‘e’ are called

A.pure logarithms
B.common logarithms
C.natural logarithms
D.infinite logarithms

Answer: C

A logarithm to base \( e \), where \( e \approx 2.71828 \), is a natural logarithm, written \( \ln x \) or \( \log_e x \).

Natural - base \( e \), the base at which the derivative of \( \log_b x \) is exactly \( 1/x \), which is why calculus uses it throughout.
Common - base 10, the one meant by a bare \( \log \) in most arithmetic work.

There is no such thing as a pure or an infinite logarithm; both are invented distractors.

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12.

Idea of logarithms was firstly introduced by

A.Henry Briggs
B.John Napier
C.Abu M. Musa Al Khwarizmi
D.Jobst Burgi

Answer: C

The credit here divides between two people, and the question asks for the idea rather than the tables.

Al-Khwarizmi (9th century) set out the underlying idea, relating a geometric progression to the arithmetic progression of its exponents.
John Napier (1614) built on that to publish the first logarithm tables, which is why he is named as the inventor of logarithms as a calculating device.
Henry Briggs then recast Napier's system to base 10, giving the common logarithms.

Read the wording carefully: introduced the idea points to Al-Khwarizmi, whereas invented logarithms would point to Napier.

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13.

Characteristic of log 1563.4 is

Answer: B

For a number of one or more digits before the decimal point, the characteristic is one less than that count of digits:

\[ \text{characteristic} \;=\; (\text{digits before the point}) - 1 \]

1563.4 has four digits before the point, so

\[ \text{characteristic} \;=\; 4 - 1 \;=\; 3 \]

This follows from \( 1563.4 = 1.5634 \times 10^{3} \), giving \( \log 1563.4 = 3 + \log 1.5634 \). The 3 is the characteristic and the \( \log 1.5634 \) part, always between 0 and 1, is the mantissa.

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14.

Characteristic of log 0.03 is

A.−1
B.−2
C.1
D.2

Answer: B

For a number smaller than 1 the characteristic is negative, and its size is one more than the count of zeros lying between the decimal point and the first significant figure:

\[ \text{characteristic} \;=\; -\left(\text{zeros after the point} + 1\right) \]

In 0.03 there is one such zero, so

\[ \text{characteristic} \;=\; -(1+1) \;=\; -2 \]

Check it against the standard form: \( 0.03 = 3 \times 10^{-2} \), so \( \log 0.03 = -2 + \log 3 \). Answer \( -1 \) is the trap for counting the zero but forgetting the extra one.

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15.

Logarithms having base ‘10’ are called

A.pure logarithms
B.common logarithms/Briggesian logarithms
C.natural logarithms
D.infinite logarithms

Answer: B

Logarithms to base 10 are the common logarithms, also called Briggsian after Henry Briggs, who computed the first base-10 tables.

They are the convenient base for hand arithmetic because our numerals are base 10: \( \log 1000 = 3 \) exactly, and multiplying a number by 10 simply adds 1 to its logarithm, which is what makes the characteristic and mantissa split work.

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16.

A concise, precise and convenient method to write very small or large numbers is called

A.equation
B.scientific notation
C.matrix
D.iota

Answer: B

Scientific notation writes a number as a figure between 1 and 10 multiplied by a power of ten, \( a \times 10^{n} \) with \( 1 \le a < 10 \).

\[ 0.000581 \;=\; 5.81 \times 10^{-4}, \qquad 1563.4 \;=\; 1.5634 \times 10^{3} \]

It is exactly this form that gives a logarithm its two parts: the exponent becomes the characteristic and the leading figure supplies the mantissa.

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17.

Number whose logarithm is given is called

A.geometric logarithm
B.anti logarithm
C.common logarithm
D.natural logarithm

Answer: B

The antilogarithm is the number whose logarithm is given, so it undoes the logarithm:

\[ \log x = y \quad\Longleftrightarrow\quad x = \operatorname{antilog} y = 10^{\,y} \]

For instance \( \log 100 = 2 \), so \( \operatorname{antilog} 2 = 100 \). Common and natural logarithms name the base; antilogarithm names the inverse operation, which is why the other options do not fit.

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18.

Napier's logarithm is also called

A.natural logarithm
B.anti logarithm
C.common logarithm
D.mantissa

Answer: A

Napier's logarithms are the natural logarithms, taken to base \( e \) and written \( \ln x \).

Common logarithms, to base 10, came slightly later and are named after Briggs. The mantissa is only the fractional part of a logarithm, and the antilogarithm is the inverse operation, so neither names a system of logarithms at all.

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19.

Characteristic of log 0.000581 is

A.3
B.4
C.5
D.-4

Answer: D

The number is below 1, so the characteristic is negative and its size is one more than the count of zeros between the decimal point and the first significant figure:

\[ \text{characteristic} \;=\; -\left(\text{zeros after the point} + 1\right) \]

0.000581 carries three such zeros, so

\[ \text{characteristic} \;=\; -(3+1) \;=\; -4 \]

Confirm it in standard form: \( 0.000581 = 5.81 \times 10^{-4} \), hence \( \log 0.000581 = -4 + \log 5.81 \). The positive options are the trap for dropping the sign that every number below 1 must carry.

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20.

Decadic logarithm is also known as

A.natural logarithm
B.anti logarithm
C.common logarithm
D.mantissa

Answer: C

The decadic logarithm is the logarithm to base 10, and its usual English name is the common logarithm, written \( \log_{10} \) or simply \( \log \).

The natural logarithm takes base \( e \approx 2.718 \) and is written \( \ln \). "Decadic" comes from the Greek for ten, which is the clue to the base meant.

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