Logarithms Questions and Answers

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.

Answer: B

Start from Boyle's law at constant temperature:

\[ V = \frac{c}{p} \]

Differentiating with respect to \( p \):

\[ \frac{dV}{dp} = -\frac{c}{p^{2}} \]

Since \( c = pV \), this becomes

\[ \frac{dV}{dp} = -\frac{pV}{p^{2}} = -\frac{V}{p} \]

The negative sign reflects that volume falls as pressure rises.

Answer: D

\[ \log(ab) = \log a + \log b \]

The logarithm of a product is the sum of the logarithms. The companion rules are

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

Turning multiplication into addition is exactly what made logarithms a calculating tool.

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.

Answer: B

A partial differential equation involves partial derivatives with respect to two or more independent variables.

\[ \frac{\partial u}{\partial t} = c^{2}\frac{\partial^{2} u}{\partial x^{2}} \]

An ordinary differential equation has just one independent variable and so no partial derivatives at all.

Answer: C

An interval that includes one endpoint but not the other is half closed (equally, half open).

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

A closed interval includes both ends and an open one neither.

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: 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.

Answer: A

Faà di Bruno's formula generalises the chain rule to higher derivatives, giving \( \tfrac{d^{n}}{dx^{n}}f(g(x)) \) as a sum over the partitions of \( n \).

The first case is the ordinary chain rule; the second is

\[ (f \circ g)'' = f''(g)\,(g')^{2} + f'(g)\,g'' \]

Leibniz's formula is the analogous generalisation for the product rule, not the chain rule.

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.