Chapter 3.3, Problem 88E

To determine

To find: the logarithmic equation that relates diameter y of a bright common wire nail to its length x .

Answer to Problem 88E

  $\ln y=0.66\ln x-2.64$

Explanation of Solution

Given:

The table:

x	2	3	4	5	6
y	0.113	0.148	0.192	0.225	0.262

Concept Used:

Slope of a line:

The slope of line passing through two point $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by $m=\frac{y_2-y_1}{x_2-x_1}$ , $x_1\ne x_2$

Slope intercept form of equation of line is $y=mx+b$ , where m is the slope of the line and $\left(0,b\right)$ is the y -intercept of the line.

Construct a new table by taking the natural logarithms of the x and y values in the given table.

$\ln x$	0.693	1.099	1.386	1.609	1.792
$\ln y$	-2.180	-1.910	-1.650	-1.492	-1.339

Now plot each point $\left(\ln x,\ln y\right)$ to make sure they lie on a line.

  

Now, choose any two points to determine the slope of the line.

So, using the points $\left(0.693,-2.180\right)$ and $\left(1.099,-1.910\right)$ , the slope of the line is:

  $\begin{array}{c}m=\frac{y{2}_{}-y_1}{x_2-x_1}\\ =\frac{-1.910-\left(-2.180\right)}{1.099-0.693}\\ =\frac{0.27}{0.406}\\ \approx 0.66\end{array}$

Now, from the slope-intercept form of line, the equation of line is:

  $Y=mX+b$

Where $Y=\ln y$ and $x=\ln x$

That is, equation of line is,

  $\ln y=0.66\ln x+b$

Now, to find b , substitute any point $\left(x,y\right)$ from the original table, so let’s substitute $\left(2,0.113\right)$ ,

  $\begin{array}{c}\ln \left(0.113\right)=0.66\ln \left(2\right)+b\\ -2.18=0.66\left(0.693\right)+b\\ b=-2.18-0.66\left(0.693\right)\\ b\approx -2.64\end{array}$

Thus, the equation of line is:

  $\ln y=0.66\ln x-2.64$

Conclusion:

The logarithmic equation that relates y to x is $\ln y=0.66\ln x-2.64$ .