Logarithms Questions and Answers

Answer: C

The credit here divides between two people, and the question asks for the idea rather than the tables.

• Al-Khwarizmi (9th century) set out the underlying idea, relating a geometric progression to the arithmetic progression of its exponents.
• John Napier (1614) built on that to publish the first logarithm tables, which is why he is named as the inventor of logarithms as a calculating device.
• Henry Briggs then recast Napier's system to base 10, giving the common logarithms.

Read the wording carefully: introduced the idea points to Al-Khwarizmi, whereas invented logarithms would point to Napier.

Answer: B

\( F(x) = -c/b \) is a non homogeneous solution - a particular solution belonging to an equation whose right-hand side is not zero.

For \( by + c = 0 \) the constant \( y = -c/b \) satisfies the equation, and it exists only because of the constant term \( c \). Remove that term and the equation is homogeneous, whose solution would be \( y = 0 \).

The general solution of a non homogeneous linear equation is the sum of the two:

\[ y \;=\; y_{\text{homogeneous}} + y_{\text{particular}} \]

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.