Logarithms Questions and Answers

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.

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: A

A linear differential equation is homogeneous when every term carries the unknown function or one of its derivatives, so the right-hand side is zero:

\[ a_n y^{(n)} + \cdots + a_1 y' + a_0 y \;=\; 0 \]

Add a term that does not involve \( y \) and it becomes non homogeneous:

\[ a_n y^{(n)} + \cdots + a_0 y \;=\; f(x) \]

The distinction matters because a homogeneous equation's solutions form a vector space: any combination of solutions is itself a solution, which is false once \( f(x) \) is present.

Answer: A

An ordinary differential equation (ODE) involves derivatives with respect to a single independent variable.

\[ \frac{d^{2}y}{dx^{2}} + 3\frac{dy}{dx} + 2y = 0 \]

When two or more independent variables appear and partial derivatives are involved, the equation is a PDE instead.

Answer: C

A real number is precisely a quantity that can be placed on the number line - every point on the line corresponds to one real number and vice versa.

Natural and whole numbers occupy only isolated points on that line, leaving gaps for the fractions and irrationals between them.

Answer: A

For a number greater than 1, the characteristic is one less than the number of digits before the decimal point.

\[ \text{characteristic} = (\text{digits in integral part}) - 1 = 3 - 1 = 2 \]

For example \( \log 342 = 2.5340 \), and 342 has three digits. This works because \( 100 \leq 342 < 1000 \), so the logarithm lies between 2 and 3.

Answer: B

Apply the chain rule to \( g(x) = \sin(2x^{2}+x) \).

\[ \frac{d}{dx}\sin u = \cos u \cdot \frac{du}{dx} \]

With \( u = 2x^{2}+x \):

\[ \frac{du}{dx} = 4x + 1 \]

\[ g'(x) = (4x+1)\cos(2x^{2}+x) \]

The outer function changes from sine to cosine; leaving it as a sine is the error the distractors are built on.

Answer: B

Cross-multiplying the proportion clears both denominators:

\[ \frac{a}{b} = \frac{c}{d} \quad\Longrightarrow\quad ad = bc \]

Multiply both sides by \( bd \): the left gives \( ad \) and the right \( bc \). This holds provided \( b \) and \( d \) are non-zero.

Answer: C

The whole point of the transform is that the subsidiary equation is algebraic: derivatives in \( t \) have become powers of \( s \).

\[ (s^{2} + as + b)Y(s) = G(s) \quad\Longrightarrow\quad Y(s) = \frac{G(s)}{s^{2}+as+b} \]

Solving it needs only algebraic manipulation - no integration or differentiation at all. The calculus returns only in the inverse transform.