So far, we have always used base 10 for a logarithmic transformation. The reason for this is that our number system is based on base 10 and it is therefore easy to logarithmically transform numbers of the form . . . , 0.01, 0.1, 1, 10, 100, 1000, . . . when we use base 10. Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship in a coordinate system whose axes are accordingly transformed so that a straight line results. $y=3^{x};$ base 3

Use the linearity property and Table of the Laplace transform to determine: (a) $\mathcal{L}\left\{\cosh ^{2} b t-\sinh ^{2} b t\right\}$ for $s>|b|$; (b) $\mathcal{L}\left\{t^{2} \cosh b t\right\}$ for $s>|b|$

Show that the limit leads to an indeterminate form. Then carry out the two-step procedure: Transform the function algebraically and evaluate using continuity.

$$\lim _ { t \rightarrow 9 } \frac { 2 t - 18 } { 5 t - 45 }$$

Use the integral definition to calculate the Laplace transform of the function

$$f(t)=\left\{\begin{array}{ll}{t-1} & {\text { if } 0 \leq t<2} \\ {1} & {\text { if } t \geq 2}\end{array}\right.$$

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Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions. Where appropriate, write f in terms of unit step functions. $y^{\prime \prime}+16 y=1, \quad y(0)=1, y^{\prime}(0)=2$

Consider the signal

$$x(t)=e^{-5t} u(t)+e^{-βt} u(t)$$

, and denote its Laplace transform by X(s). What are the constructs placed on the real and imaginary parts of β if the region of convergence of X(s) is Re(s)>-3?

The formula for converting kilometers to miles is m = 1.609k, where m is miles and k is kilometers. Transform this formula to find a formula for converting miles to kilometers. Use this formula to determine the number of kilometers in 539 miles and in 7380 miles, rounded to the nearest kilometer.

Transform the given initial value problem

$$\begin{gather*} t \ U'' + U' + t \ U = 0, \,\,\, U(1) = 1, \,\,\, U'(1) = 0 \end{gather*}$$

into an initial value problem for the two first order equations. Then write the system in matrix form.

Find the Laplace transform of the given function

$$\begin{gather*} \begin{cases} 0, & 0 \leq t \leq 1, \\ 1, & 1 < t \leq 2 \\ 0, & 2 < t. \end{cases} \end{gather*}$$

describe how to transform the graph of an appropriate monomial function

$$f(x)=x^n$$

into the graph of the given polynomial function. Sketch the transformed graph by hand and support your answer with a grapher. Compute the location of the y-intercept as a check on the transformed graph.

$$g(x)=-(x+5)^3$$

Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. $y^{\prime \prime}+9 y=\cos 3 t, \quad y(0)=2, \quad y^{\prime}(0)=5$

Find the Laplace transform of $f(t).$

$$f(t)=\left\{\begin{array}{ll}{1-\cos (2 t)} & {\text { for } 0 \leq t<3 \pi} \\ {0} & {\text { for } t \geq 3 \pi}\end{array}\right.$$

Sketch or graph the given function, which is assumed to be zero outside the given interval. Represent it, using unit step functions. Find its transform. Show the details of your work. sinh t(0<t<2)

Sketch or graph the given function (which is assumed to be zero outside the given interval). Represent it using unit step functions. Find its transform. Show the details of your work.

$e^t(0<t<2)$