Logarithms Questions and Answers

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: B

Linearity is judged in the dependent variable and its derivatives: they may appear only to the first power and never multiplied together.

\[ (y-x)\,\partial x + 4xy\,\partial y = 0 \]

Rearranged as

\[ \frac{dy}{dx} = \frac{x-y}{4xy} \]

the term \( 4xy\,\tfrac{dy}{dx} \) has \( y \) multiplying its own derivative, so the equation is non-linear.

A linear first-order equation has the form \( \tfrac{dy}{dx} + P(x)y = Q(x) \), with no product of \( y \) and \( y' \).

Answer: B

A rate of change is a derivative, and an equation containing derivatives is a differential equation.

\[ \frac{dy}{dt} = f(t, y) \]

This is why differential equations model so much of physics, chemistry and biology - those subjects are largely statements about how things change.

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: C

The parenthesis at \( b \) excludes that endpoint, so the interval is open; it extends to \( +\infty \) with no upper endpoint, so it is unbounded.

\[ (b, +\infty) = \{\, x \mid x > b \,\} \]

Hence open and unbounded. Infinity is never an included endpoint, so that side always carries a parenthesis.

Answer: D

Square brackets include the endpoint; round brackets exclude it.

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

So \( [a,b] \) is the closed interval. \( (a,b) \) is open, and the two mixed forms are half-open.

Answer: A

A homogeneous system \( AX = 0 \) always has the trivial solution \( X = 0 \). It has only that solution when \( A \) is invertible, which happens exactly when

\[ \det A \neq 0 \]

If the determinant were zero, the matrix would be singular and infinitely many non-trivial solutions would exist.

Answer: C

The quotient rule turns a division into a subtraction:

\[ \ln\frac{a}{b} \;=\; \ln a - \ln b \]

It follows from the product rule. Writing \( a = \frac{a}{b}\cdot b \) gives \( \ln a = \ln\frac{a}{b} + \ln b \), and rearranging leaves the result.

Note what it is not: \( \ln\frac{a}{b} \) is not \( \frac{\ln a}{\ln b} \). That last expression is the change-of-base ratio, a different quantity altogether.

Answer: B

Base \( e \) gives the natural logarithm, written \( \ln x \) or \( \log_e x \), where

\[ e \;=\; \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^{n} \;\approx\; 2.718281828459 \]

It is called natural because it is the base at which the logarithm's own rate of change is simply \( 1/x \), with no stray constant:

\[ \frac{d}{dx}\ln x \;=\; \frac{1}{x} \]

Any other base introduces a factor of \( 1/\ln b \), which is why calculus uses this one throughout.

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.