Chapter 3.3, Problem 94E

To determine

To find: The logarithmic equation that relates the diameter of nail, $y$ and length of nail, $x$ .

Answer to Problem 94E

  $\ln y=\frac{1}{2}\ln x$ ; wherediameter of nail is $y$ and length of nail is $x$ .

Explanation of Solution

Given information:

A table of values is given as −

  

Calculation:

A table of values is given as −

  

Step $1$ : Taking natural logarithm of both $x\text{ and }y$ .

$\ln x$	$0$	$0.693$	$1.0986$	$1.3863$	$1.6094$
$\ln y$	$-2.631$	$-2.120$	$-1.910$	$-1.595$	$-1.435$

Step $2$ : Plot these points on a coordinate axes.

  

All points are in straight line.

Step $3$ : Take any two points to determine slope of the line.

  $m=\frac{-2.120-(-2.631)}{0.693-0}=\frac{1}{2}$

Step $4$ : Using point slope formula, equation of the line would be -

  $\begin{array}{l}Y=mX\\ \Rightarrow Y=\frac{1}{2}X\text{ ;where }Y=\ln y,X=\ln x\\ \Rightarrow \ln y=\frac{1}{2}\ln x\end{array}$

Hence, desired logarithmic equation is $\ln y=\frac{1}{2}\ln x$ .