Logarithms Questions and Answers

Answer: A

A variable is a quantity that can change during an experiment - the thing being varied or measured.

A function is a rule relating variables, and an equation a statement of equality between expressions; neither is the changing quantity itself.

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: 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

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.

Answer: D

A continuous model treats its quantities as varying smoothly over the whole domain, with a value defined at every point - a velocity field, a temperature distribution, an electric field.

A discrete model would define values only at separated points or time steps. The distinction is what separates differential equations from difference equations.

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.

Answer: C

A stochastic model includes randomness, so its outcomes are described by probability distributions rather than fixed values. It is therefore also called a statistic (or probabilistic) model.

Its opposite is the deterministic model, where the same inputs always give the same output.

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.

Answer: A

In \( F(x, y, y', \ldots, y^{(n)}) = 0 \), the variable \( x \) is independent and \( y \) is the unknown function of \( x \).

\[ y = y(x), \qquad y' = \frac{dy}{dx} \]

The primes are derivatives with respect to \( x \), which is what fixes \( x \) as the independent variable.

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.