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Chapter 4: Problem 75

Solve the equations in parts (a)-(c) by inspection. Then estimate the solutions to parts (d) and (e) between two consecutive integers. a. \(2^{x}=4\) b. \(2^{x}=8\) c. \(2^{x}=16\) d. \(2^{x}=7\) e. \(2^{x}=10\)

Short Answer

Expert verified
a. x = 2, b. x = 3, c. x = 4, d. x 鈮� 2.8, e. x 鈮� 3.3

Step by step solution

01

Part (a): Solve the equation 2^x = 4

Recognize that 4 is a power of 2. Express 4 as 2^2. Hence, we have 2^x = 2^2 which implies x = 2.
02

Part (b): Solve the equation 2^x = 8

Recognize that 8 is a power of 2. Express 8 as 2^3. Thus, we have 2^x = 2^3 which implies x = 3.
03

Part (c): Solve the equation 2^x = 16

Recognize that 16 is a power of 2. Express 16 as 2^4. Therefore, we have 2^x = 2^4 which implies x = 4.
04

Part (d): Estimate the solution for 2^x = 7

Recognize that 7 is between 4 (which is 2^2) and 8 (which is 2^3). Therefore, x is between 2 and 3. Since 7 is closer to 8, x is closer to 3. Estimate x 鈮� 2.8.
05

Part (e): Estimate the solution for 2^x = 10

Recognize that 10 is between 8 (which is 2^3) and 16 (which is 2^4). Therefore, x is between 3 and 4. Since 10 is closer to 8, x is closer to 3. Estimate x 鈮� 3.3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Exponentials
Solving exponential equations involves finding the variable in the exponent. To solve equations like these, we first recognize that the unknown variable is in the power of a given base, usually a number like 2, 3, or 10. For example, for the equation \(2^x = 8\), we need to determine the value of \(x\) that makes the equation true. Recognizing patterns is critical. We know that \(8 = 2^3\), so by matching the exponents, we find \(x = 3\).

When solving exponentials, it's helpful to:
  • Recognize powers of the base number involved (like \(2^2 = 4\), \(2^3 = 8\), \(2^4 = 16\), etc.).
  • Rewrite both sides of the equation as powers of the same base, whenever possible.
This method allows us to directly compare and solve for the unknown exponent.
Estimation in Algebra
When an exact solution isn't apparent in an exponential equation, estimation helps us find an approximate answer. For instance, consider \(2^x = 7\). We know \(7\) lies between \(4\) (which is \(2^2\)) and \(8\) (which is \(2^3\)), so \(x\) must be between \(2\) and \(3\).

By comparing the distances, we can estimate \(x\). Since \(7\) is closer to \(8\) than \(4\), a reasonable estimate for \(x\) would be around \(2.8\). Similarly, for \(2^x = 10\), \(10\) falls between \(8\) (\(2^3\)) and \(16\) (\(2^4\)), giving \(x\) a range between \(3\) and \(4\). Because \(10\) is closer to \(8\), \(x\) might be around \(3.3\).
  • Estimation allows us to narrow down the range for \(x\).
  • It involves locating the target value within known power intervals and judging the closeness.
Powers of Two
Understanding powers of two is crucial for solving and estimating these equations. Powers of two are numbers like 2, 4, 8, 16, 32, and so on, where each subsequent number is double the previous one.

Let's highlight some important points:
  • \(2^1 = 2\)
  • \(2^2 = 4\)
  • \(2^3 = 8\)
  • \(2^4 = 16\)
These relationships help us quickly identify the exponential form of a number. For example, recognizing that \(16\) is \(2^4\) helps us immediately determine that in the equation \(2^x = 16\), \(x\) must be \(4\).

Familiarity with these powers will speed up your problem-solving process in algebra.
Consecutive Integers
Consecutive integers are integers that follow each other in order without gaps. In the context of exponential equations, understanding consecutive integers helps with estimating solutions.

For example, when dealing with \(2^x = 7\), we observe that \(7\) lies between \(4\) (\(2^2\)) and \(8\) (\(2^3\)). Recognizing these consecutive powers of two allows us to see that \(x\) must lie between the consecutive integers \(2\) and \(3\).
  • \(x\) is more specifically estimated by evaluating how close \(7\) is to \(4\) and \(8\).
  • In general, finding which two powers a number lies between helps us narrow the possible values of \(x\).
This method aids in developing an intuitive understanding of the relationships between different powers and their corresponding exponents.

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Most popular questions from this chapter

A logarithmic function \(y=\log _{b} x\) with base \(b>1\) increases over its domain. However, the rate of increase decreases with larger and larger values of \(x .\) For Exercises \(119-120\), demonstrate this statement by finding the average rate of change on each interval \([a, b] .\) Round to 4 decimal places where necessary. \(\mathrm{Y}_{1}=\ln x\) a. [0.5,1] b. [1,10] c. [10,20] d. [20,30]

\(F(C)=\frac{9}{5} C+32\) gives the temperature in degrees Fahrenheit as a function of the temperature \(C\) in degrees Celsius. Find an equation for \(C(F)\) and interpret its meaning in the context of this problem.

Determine if the statement is true or false. The domain of any one-to-one function is the same as the domain of its inverse function.

According to the CIA's World Fact Book, in \(2010,\) the population of the United States was approximately 310 million with a \(0.97 \%\) annual growth rate. (Source: www.cia.gov) At this rate, the population \(P(t)\) (in millions) can be approximated by \(P(t)=310(1.0097)^{t}\), where \(t\) is the time in years since 2010 . a. Is the graph of \(P\) an increasing or decreasing exponential function? b. Evaluate \(P(0)\) and interpret its meaning in the context of this problem. c. Evaluate \(P(10)\) and interpret its meaning in the context of this problem. Round the population value to the nearest million. d. Evaluate \(P(20)\) and \(P(30)\). e. Evaluate \(P(200)\) and use this result to determine if it is reasonable to expect this model to continue indefinitely.

Fluorodeoxyglucose is a derivative of glucose that contains the radionuclide fluorine- \(18\left({ }^{18} \mathrm{~F}\right) .\) A patient is given a sample of this material containing \(300 \mathrm{MBq}\) of \({ }^{18} \mathrm{~F}\) (a megabecquerel is a unit of radioactivity). The patient then undergoes a PET scan (positron emission tomography) to detect areas of metabolic activity indicative of cancer. After \(174 \mathrm{~min}\), one-third of the original dose remains in the body. a. Write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the radioactivity level \(Q(t)\) of fluorine- 18 at a time \(t\) minutes after the initial dose. b. What is the half-life of \({ }^{18} \mathrm{~F}\) ?
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