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Chapter 7: Problem 7

In Exercises 7-8, a. Rewrite each equation in exponential form. b. Use a table of coordinates and the exponential form from part (a) to graph each logarithmic function. Begin by selecting \(-2,-1,0,1\), and 2 for \(y\). \(y=\log _{4} x\)

Short Answer

Expert verified
The exponential form is \(4^y = x\). The coordinates for the graph are (-2, 0.0625), (-1, 0.25), (0, 1), (1, 4), (2,16).

Step by step solution

01

Convert logarithm to exponential form

To convert \(y=\log _{4} x\) to exponential form, recall that logarithms and exponentials are two sides of the same coin. Using the rule \(y = \log_b a\) is equivalent to \(b^y = a\), the exponential form of the given equation is \(4^y = x\).
02

Create table of coordinates

Based on the exponential form, evaluate the expression \(x = 4^y\) for \(y = -2, -1, 0, 1, 2\). This will provide the x-coordinates:\(y = -2\) yields \(x = 4^-2 = 0.0625\)\(y = -1\) yields \(x = 4^-1 = 0.25\)\(y = 0\) yields \(x = 4^0 = 1\)\(y = 1\) yields \(x = 4^1 = 4\)\(y = 2\) yields \(x = 4^2 = 16\)
03

Plot the graph

To plot the graph, make a Cartesian plane, mark the x and y axes, and plot the following points: (-2, 0.0625), (-1, 0.25), (0, 1), (1, 4), (2,16). Draw a curve that passes through these points. Note that the curve will never touch or cross the y-axis, due to the nature of logarithmic functions.

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Most popular questions from this chapter

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, \(y\), in feet, in terms of its horizontal distance, \(x\), in feet. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \boldsymbol{x} \text {, Ball's } \\ \text { Horizontal } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \boldsymbol{y}, \\ \text { Ball's } \\ \text { Height } \\ \text { (feet) } \end{array} \\ \hline 0 & 6 \\ \hline 1 & 7.6 \\ \hline 3 & 6 \\ \hline 4 & 2.8 \\ \hline \end{array} $$ $$ \begin{aligned} &\text { [adadReg }\\\ &\begin{aligned} &y=3 \times 2+b x+c \\ &\bar{y}=.8 \\ &b=2.4 \\ &c=6 \end{aligned} \end{aligned} $$ a. Explain why a quadratic function was used to model the data. Why is the value of \(a\) negative? b. Use the graphing calculator screen to express the model in function notation. c. Use the model from part (b) to determine the \(x\)-coordinate of the quadratic function's vertex. Then complete this statement: The maximum height of the ball occurs ____ feet from where it was thrown and the maximum height is ____ feet.

In Exercises 27-28, use the directions for Exercises 9-14 to graph each quadratic function. Use the quadratic formula to find \(x\)-intercepts, rounded to the nearest tenth. \(f(x)=-2 x^{2}+4 x+5\)

What is an exponential function?

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{r}2 x-y<3 \\ x+y<6\end{array}\right.\)

Many elevators have a capacity of 2000 pounds. a. If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when \(x\) children and \(y\) adults will cause the elevator to be overloaded. b. Graph the inequality. Because \(x\) and \(y\) must be positive, limit the graph to quadrant I only. c. Select an ordered pair satisfying the inequality. What are its coordinates and what do they represent in this situation?
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