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Chapter 4: Problem 137

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

Short Answer

Expert verified
Yes, the statement makes sense. Despite the fact that the values on the right hand side of the equations are different, the equations are similar and can be solved using the same method, which is typically done using logarithms.

Step by step solution

01

Analyze the equations

By examining both equations, it can be seen that they have the same base on the left hand side - which is 2. Furthermore, the exponential variable is also the same in both equations. Therefore, both equations do have similar structural format.
02

Evaluate the reasoning

The given reasoning is that since the equations are similar, they are solved using the same method. This reasoning is valid because irrespective of the difference in right hand side values, the general method of solving exponential equations remains the same - that is, by applying logarithmic operations.
03

Conclusion

The statement given in the question does make sense. Although the equations are not identical, they are similar as they follow the same equation structure and hence can be solved using the same method.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Examples of exponential equations include \(10^{x}=5.71\) \(e^{x}=0.72,\) and \(x^{10}=5.71\)

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(y=2+\log _{3} x, y=\log _{3}(x+2),\) and \(y=-\log _{3} x\) in the same viewing rectangle as \(y=\log _{3} x .\) Then describe the change or changes that need to be made to the graph of \(y=\log _{3} x\) to obtain each of these three graphs.

Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled by exponential or logarithmic functions. Group members should select the two sets of data that are most interesting and relevant. For each set selected, find a model that best fits the data. Each group member should make one prediction based on the model and then discuss a consequence of this prediction. What factors might change the accuracy of the prediction?

In Example I on page \(520,\) we used two data points and an exponential function to model the population of the United States from 1970 through 2010 . The data are shown again in the table. Use all five data points to solve Exercises \(66-70\). $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { U.S. Population }} \\ {\text { after } 1969} & {\text { (millions) }} \\ {1(1970)} & {203.3} \\ {11(1980)} & {226.5} \\ {21(1990)} & {248.7} \\ {31(2000)} & {281.4} \\ {41(2010)} & {308.7} \end{array} $$ Use the values of \(r\) in Exercises \(66-69\) to select the two model= of best fit. Use each of these models to predict by which yeathe U.S. population will reach 335 million. How do these answers compare to the year we found in Example \(1,\) namel \(=\) \(2020 ?\) If you obtained different years, how do you account fo this difference?

Exercises 150鈥�152 will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 32\) b. Evaluate: \(\log _{2} 8+\log _{2} 4\) c. What can you conclude about \(\log _{2} 32,\) or \(\log _{2}(8 \cdot 4) ?\)
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