Logarithms Questions and Answers

Answer: A

A common logarithm splits into a whole-number characteristic, fixed by the position of the decimal point, and a fractional mantissa, fixed by the digits themselves.

\[ \log 3.42 = 0.5340, \quad \log 34.2 = 1.5340, \quad \log 342 = 2.5340 \]

The digits 342 give the same mantissa 0.5340 every time; only the characteristic changes. This is exactly why one page of log tables covers every power of ten.

Answer: C

There are two logarithms in ordinary use:

• Common, to base 10, written \( \log \)
• Natural, to base \( e \), written \( \ln \)

Any base is mathematically possible, but these two are the ones tabulated and built into calculators.

Answer: C

A logarithm table has three parts: the column of leading digits down the left, the main body giving the mantissa for the first three significant digits, and the smaller mean difference columns on the right that adjust for a fourth digit.

The characteristic is not tabulated at all - it is read off from the position of the decimal point.

Answer: A

Logarithms turn multiplication into addition and division into subtraction:

\[ \log(ab) = \log a + \log b, \qquad \log\left(\frac{a}{b}\right) = \log a - \log b \]

Long products and quotients become short sums, and because the characteristic records the power of ten separately, a stray zero cannot be gained or lost. That was their whole purpose before calculators.

Answer: A

For a number greater than 1, the characteristic is one less than the number of digits before the decimal point.

\[ \text{characteristic} = (\text{digits in integral part}) - 1 = 3 - 1 = 2 \]

For example \( \log 342 = 2.5340 \), and 342 has three digits. This works because \( 100 \leq 342 < 1000 \), so the logarithm lies between 2 and 3.

Answer: C

By definition, \( \log_{a} x \) is the power to which \( a \) must be raised to give \( x \). Since

\[ a^{1} = a \]

it follows that

\[ \log_{a} a = 1 \]

for every valid base. The companion result is \( \log_{a} 1 = 0 \), since \( a^{0} = 1 \).

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.

Answer: D

The fractional part of a common logarithm is the mantissa; the whole-number part is the characteristic.

\[ \log 342 = 2.5340 \quad\Longrightarrow\quad \text{characteristic } 2, \ \text{mantissa } 0.5340 \]

The mantissa depends only on the sequence of digits, which is why log tables need list it just once for each digit string.

Answer: A

The mean Earth–Sun distance is about \( 1.5 \times 10^{8} \) km - 150,000,000 km.

This distance is itself the definition of the astronomical unit (AU), the standard yardstick for distances within the solar system. Light covers it in about 8 minutes 20 seconds.

Answer: D

The characteristic is one less than the count of digits before the decimal point.

\[ \text{characteristic} = 1 - 1 = 0 \]

A one-digit number lies between 1 and 10, so its logarithm lies between \( \log 1 = 0 \) and \( \log 10 = 1 \) - the whole part must therefore be 0. For instance \( \log 7 = 0.8451 \).