Chapter 4.4, Problem 63E

To determine

To calculate: The change of base formula to show that, ${\log }_{3}x=\frac{Inx}{In3}$ use this fact to draw the graph of the function $f\left(x\right)={\log }_{3}x$ .

Answer to Problem 63E

The slope of the line is 0,1,2,3…... The ratio of the output is $x=1$ ,2,3,4….. Ratio of output to input at both the points is not same so it is not a proportional relation. As x increases the slop of graph value of the function . Graph of the function is provided below,

  

Explanation of Solution

Given information:

The function of graph $f\left(x\right)={\log }_{3}x$ ,

Formula used:

The base formula ${\log }_{3}x=\frac{Inx}{In3}$ .

Check the function at x=0,1,2,3………

Calculation:

It is provided that the equation of function is $f\left(x\right)={\log }_{3}x$ .

  $f\left(x\right)={\log }_{3}x$

Substitute the value x=1,

  $\begin{array}{l}f\left(x\right)={\log }_{3}x\\ f\left(x\right)=\frac{\log x}{\log 3}\\ f\left(1\right)=\frac{\log 1}{\begin{array}{l}\log 3\\ =\frac{0}{\log 3}\end{array}}\end{array}$

Here the result is 0.

Substitute the value x=2,

  $\begin{array}{l}f\left(x\right)=\frac{\log x}{\log 3}\\ f\left(1\right)=\frac{\log 2}{\begin{array}{l}\log 3\\ =0.630929\end{array}}\end{array}$

For the value of x=3, $\begin{array}{l}f\left(x\right)=\frac{\log x}{\log 3}\\ f\left(1\right)=\frac{\log 3}{\begin{array}{l}\log 3\\ =1\end{array}}\end{array}$

Therefore, the result is continue in this way:

So the graph of this equation is as:

For every x .the graph is straight line graph.

  

Thus, slope of the line is straight line graph for every x. The ratio of the output to the input when $x=1$ is 0 and ratio of the output to the input when $x=2$ is 0.630929. Ratio of output to input at both the points is not same so it is a graph given above. As x increases the value of the function increases .