Logarithms Questions and Answers

Answer: D

A logarithmic scale spaces values by their logarithm rather than by their size, so each fixed step multiplies the quantity by a constant factor:

\[ 1,\; 10,\; 100,\; 1000 \;\longrightarrow\; 0,\; 1,\; 2,\; 3 \]

That is what lets a single axis hold a range spanning many orders of magnitude. The decibel, the Richter scale and pH all work this way. Non linear scale is true of a logarithmic scale but describes a whole family rather than naming this one.

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

An interval is closed when both endpoints are finite and both belong to it, written with square brackets:

\[ [a,b] \;=\; \{\, x : a \le x \le b \,\} \]

• Closed \( [a,b] \) - both endpoints included.
• Open \( (a,b) \) - neither endpoint included.
• Infinite - at least one end unbounded, as in \( [a,\infty) \).

Answer: B

The notation \( \ln \) always means the logarithm to base \( e \), so \( \ln 2 \) is a natural logarithm - the power to which \( e \) must be raised to give 2:

\[ \ln 2 \;=\; \log_e 2 \;\approx\; 0.693 \]

The trap is the numeral 2 in the bracket, which suggests base 2. The 2 is the argument, not the base. A binary logarithm is written \( \log_2 x \) and takes 2 as its base; the two are different, and indeed \( \log_2 2 = 1 \) whereas \( \ln 2 \approx 0.693 \).

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: A

An ordinary differential equation involves one independent variable, so every derivative in it is an ordinary derivative:

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

A partial differential equation has two or more independent variables and therefore partial derivatives, as in \( \partial u/\partial t = \partial^{2}u/\partial x^{2} \). Linear and non linear describe how the unknown appears, which is a separate classification from how many independent variables there are.

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: C

An integer interval holds only the whole numbers lying between its endpoints, rather than every real number there:

\[ \{\, x \in \mathbb{Z} : a \le x \le b \,\} \]

So between 2 and 5 it contains 2, 3, 4 and 5, whereas the closed interval \( [2,5] \) contains every real number in that range. Open and closed describe whether the endpoints are included; this option describes which kind of number is included.

Answer: B

Separation of variables is also called the Fourier method, after Fourier's use of it on the heat equation.

The idea is to look for a solution built as a product of single-variable functions:

\[ u(x,t) \;=\; X(x)\,T(t) \]

Substituting and dividing through leaves each side depending on one variable only, so both must equal a constant. That turns one partial differential equation into two ordinary ones, whose solutions are then recombined as a Fourier series.

Answer: D

Both \( \pi \) and \( e \) are transcendental: neither is a root of any polynomial equation with integer coefficients.

An algebraic number is one that is such a root - \( \sqrt{2} \) qualifies, since it satisfies \( x^{2} - 2 = 0 \). No such equation exists for \( \pi \) or \( e \); Hermite proved it for \( e \) in 1873 and Lindemann for \( \pi \) in 1882, and that second proof is what finally settled the impossibility of squaring the circle.

Every transcendental number is irrational, but the reverse fails: \( \sqrt{2} \) is irrational and still algebraic.