Chapter 3.3, Problem 61E

Solution

(a)

To calculate: earthquake magnitudes.

Magnitude $3$ is $10$ times more powerful.

Given information: magnitudes 2 and 3.

Formula used: definition.

Calculation: an earthquake of magnitude $3$ is $10$ times of ground motion of an earthquake of magnitude 2 and an earthquake of magnitude 5 is 1000 times of ground motion of an earthquake of magnitude 2.

(b)

To calculate: scatter of plot of data.

The graph is exponential

Given information: from table.

Formula used: graph table.

Calculation: make a scatter plot as shown. Then y-axis will go to $1000000$ making the first three points appear to be close to x-axis as they are much smaller.

  

(c)

To calculate: exponential of data.

  $y={10}^x$

Given information: magnitudes 2 and 3.

Formula used: definition.

Calculation: an oblivious exponential function is $y$ as function of x and to use a base of $10$ since y-values are powers of 10:

  $y={10}^x$

(d)

To calculate: logarithmic function.

  $x=\log y$

Given information: magnitudes 2 and 3.

Formula used: definition.

Calculation: model with a logarithmic function, we must have x as a function of y. since base is $10$ , we simply use log:

  $x=\log y$

(e)

To calculate: advantages logarithmic function.

  $x=\log y$

Given information: magnitudes 2 and 3.

Formula used: definition.

Calculation: some possible advantages will display of wider range of data is possible and increase in data for smaller values (magnitude).

(f)

To calculate: are function (c) and (d) inverse functions.

No

Given information: magnitudes 2 and 3.

Formula used: definition.

Calculation: no, since we use the since data for same variables, they basically are same function. The function in part (c) is y as a function of x while function in part (c) is x as a function of y.