Chapter 4.4, Problem 64E

To determine

To calculate: The family of function, $y={\log }_{a}x$ for $a=2,e,5,10$ on the same screen,use the viewing rectangular $\left[0,5\right]by\left[-3,3\right]$ .how the graphs relates.

Answer to Problem 64E

It is a positive real number graph of the exponential function with the base a passes the horizontal line test .in a particular the natural logarithm function with base e ${\log }_{a}x=1$ , $\begin{array}{c}x=\frac{In\left(x\right)}{In\left(a\right)}\\ In\left(x\right)=\left[\frac{1}{In\left(a\right)\cdot \frac{1}{x}}\right]\\ =\frac{1}{xIn\left(a\right)}\end{array}$

  

Explanation of Solution

Given information:

The family of function, $y={\log }_{a}x$ for $a=2,e,5,10$ ,

Formula used:

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

Check the function at x=2,e,5,10........

Calculation:

It is provided that the equation of function is $y={\log }_{a}x$ .

  $y={\log }_{a}x$

Substitute the value a=2 and x also being a real no.

  $\begin{array}{l}y={\log }_{a}x\\ y=\frac{\log x}{\log a}\\ y=\frac{\log 1}{\log 2}\\ y=0.5213902277\end{array}$

Here the result is 1.

Substitute the value x=2,

  $\begin{array}{l}y=\frac{\log 2}{\log e}\\ y=\frac{\log 2}{\begin{array}{c}1\\ =0.3010299957\end{array}}\end{array}$

For the value of x=3

  $\begin{array}{l}y=\frac{\log x}{\log 5}\\ y=\frac{\log 3}{\begin{array}{l}\log 5\\ =0.6826061945\end{array}}\end{array}$

Therefore, the result is continue in this way:

So the graph of this equation is as:

For every x .the graph is cruve shaped graph.

  

Thus, curve of the line for every a is a positive number except a=1 is a graph of exponential function with the base a passes the horizontal line test .In this case .