Logarithms Questions and Answers

Answer: B

The elementary functions - polynomials, exponentials, logarithms, trigonometric functions and their inverses - are closed under arithmetic operations and composition: adding, subtracting, multiplying, dividing or composing two of them always gives another elementary function.

They are not closed under integration, which is why \( \int e^{-x^{2}}dx \) escapes the class altogether.

Answer: A

The product rule for logarithms holds in any base, including base \( e \).

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

The companion rules are \( \ln\tfrac{a}{b} = \ln a - \ln b \) and \( \ln a^{n} = n\ln a \). A product inside becomes a sum outside - never a product of the two logarithms.

Answer: A

The logarithm to base 2 is the binary logarithm, written \( \log_{2} \) or \( \text{lb} \).

\[ \log_{2} 8 = 3 \quad\text{since}\quad 2^{3} = 8 \]

It is the natural measure in computing, where it counts bits and gives the depth of a balanced binary tree.

Answer: C

The logarithm to base 10 is the common logarithm, written \( \log_{10} \) or simply \( \log \).

\[ \log 1000 = 3 \quad\text{since}\quad 10^{3} = 1000 \]

It is the one tabulated in log tables, because our number system is decimal and the characteristic then records the power of ten directly.

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: D

A function has a Laplace transform only if the integral

\[ \int_{0}^{\infty} e^{-st}f(t)\,dt \]

converges. That requires \( f \) to be piecewise continuous and of exponential order.

\( \tan t \) has infinite discontinuities at \( t = \tfrac{\pi}{2}, \tfrac{3\pi}{2}, \ldots \), where it blows up faster than any exponential can damp. The integral diverges, so no Laplace transform exists.

Answer: B

A differential equation whose right-hand side does not contain the independent variable explicitly is called autonomous.

\[ \frac{dy}{dx} = f(y) \quad\text{is autonomous}; \qquad \frac{dy}{dx} = f(x,y) \quad\text{in general is not} \]

Autonomous equations are unchanged by a shift in \( x \), which is why their direction fields look the same in every vertical strip.

Answer: C

A constant factor passes straight through differentiation.

\[ \frac{\partial}{\partial x}(\epsilon x) = \epsilon \cdot \frac{\partial x}{\partial x} = \epsilon \cdot 1 = \epsilon \]

Since \( \epsilon \) does not depend on \( x \), it is not itself differentiated.

Answer: D

Volume changes at the difference between what flows in and what flows out.

\[ \frac{dV}{dt} = \text{rate in} - \text{rate out} \]

The two rates are equal and constant, so the difference is zero:

\[ \frac{dV}{dt} = 0 \]

The volume is therefore constant - the tank is in steady state, neither filling nor draining.

Answer: D

The value prescribed at the starting point of a problem is the initial condition, also called the seed value.

\[ y(t_{0}) = y_{0} \]

It is what picks a single solution out of the general family, each member of which differs only by the arbitrary constant.