Logarithms Questions and Answers

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: 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

Operational calculus replaces the operations of calculus with algebraic ones. The Laplace transform is its best-known tool: differentiation in \( t \) becomes multiplication by \( s \).

\[ \mathcal{L}\{f'(t)\} = sF(s) - f(0) \]

A differential equation therefore turns into an algebraic one, which is solved and then transformed back.

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: B

The general solution of a linear homogeneous equation is called the complementary function.

Every such equation has the trivial solution \( y = 0 \), which is why one solution is always "automatic"; the complementary function collects all of them.

For an inhomogeneous equation the full solution is the complementary function plus a particular integral.

Answer: A

A curve can be parametrised in infinitely many ways - changing the parameter traces the same set of points at a different rate.

\[ (\cos t, \sin t) \quad\text{and}\quad (\cos 2t, \sin 2t) \]

both describe the unit circle. So parametric representations are non-unique: the curve determines the shape, not the parametrisation.

Answer: D

The fractional part of a common logarithm is the mantissa; the whole-number part is the characteristic.

\[ \log 342 = 2.5340 \quad\Longrightarrow\quad \text{characteristic } 2, \ \text{mantissa } 0.5340 \]

The mantissa depends only on the sequence of digits, which is why log tables need list it just once for each digit string.

Answer: C

An ordinary differential equation has a single independent variable, so it describes a one-dimensional system - quantities evolving along one axis, usually time.

Two or more independent variables call for a partial differential equation instead.

Answer: B

An initial value problem is a differential equation together with the value of the unknown function (and enough of its derivatives) at one point:

\[ y' = f(t,y), \qquad y(t_{0}) = y_{0} \]

The equation is what makes it a differentiation problem; the initial condition merely selects one solution from the general family.

Answer: A

Transforming \( y'' - y = 0 \) with \( y(0) = 12 \) and \( y'(0) = 0 \):

\[ s^{2}Y - s \cdot 12 - 0 - Y = 0 \]

\[ Y(s^{2} - 1) = 12s \quad\Longrightarrow\quad Y = \frac{12s}{s^{2}-1} \]

Since \( \mathcal{L}\{\cosh t\} = \dfrac{s}{s^{2}-1} \):

\[ y = 12\cosh t \]

The hyperbolic cosine appears because the denominator is \( s^{2} - 1 \); an ordinary cosine would need \( s^{2} + 1 \).