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Chapter 4: Problem 150

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 \(\log _{2}(8 \cdot 4) ?\)

Short Answer

Expert verified
a. \(\log _{2} 32 = 5\). b. \(\log _{2} 8+\log _{2} 4 = 5\). c. \(\log _{2} 32 = \log _{2}(8 \cdot 4)\)

Step by step solution

01

Evaluate \(\log _{2} 32\)

This means finding the power to which the number 2 must be raised in order to get 32. We know that \(2^{5} = 32\). So, \(\log _{2} 32 = 5\).
02

Evaluate \(\log _{2} 8+\log _{2} 4\)

This requires evaluating each of the logarithmic terms separately, and then adding them together. We know that \(2^{3} = 8\) and \(2^{2} = 4\). Therefore, \(\log _{2} 8 = 3\) and \(\log _{2} 4 = 2\). The sum of these two values is \(3 + 2 = 5\)
03

Conclusion about \(\log _{2} 32\) and \(\log _{2}(8 \cdot 4)\)

Observe that the results from steps 1 and 2 are equal, both yielded 5. This confirms the logarithmic property: \(\log _{b} (a \cdot c) = \log _{b} a + \log _{b} c \). So, it can be concluded that \(\log _{2} 32 = \log _{2}(8 \cdot 4)\)

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Key Concepts

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

Logarithmic Expressions
Logarithmic expressions involve numbers and logarithms which help us find unknown exponents. A logarithm tells us which power a base number must be raised to produce a specified value. For example, considering the expression \(\log_{2} 32\), we are looking for the number we must raise 2 to in order to get 32. This kind of expression is common in both math and science as it simplifies complex calculations.
  • In \(\log_{2} 32\), the base is 2, and the result is 32.
  • The question is: "To what power must 2 be raised to equal 32?"
Since \(2^5 = 32\), the answer is 5, so \(\log_{2} 32 = 5\). Logarithms reverse the process of exponentiation, breaking down the process of multiplying the same number repeatedly quickly.
Properties of Logarithms
Logarithms have several useful properties that make them extremely helpful in simplifying expressions and solving equations. A key property used in the exercise is the Product Property of Logarithms. It states: if \(b > 0\) and \(b e 1\), then \(\log_{b} (a \cdot c) = \log_{b} a + \log_{b} c\). This property allows us to break down complex logarithmic expressions into simpler parts.
  • Evaluate \(\log_{2} 8 + \log_{2} 4\) by separately calculating each logarithm.
  • Since \(\log_{2} 8 = 3\) and \(\log_{2} 4 = 2\), the sum is \(3 + 2 = 5\).
Thus, \(\log_{2} 8 + \log_{2} 4 = \log_{2} (8 \cdot 4) = \log_{2} 32\), demonstrating how logarithms can simplify multiplication of numbers into the addition of logarithms.
Algebra
Understanding logarithms is deeply connected with the principles of algebra. Algebra helps us see the relationships between numbers and the operations that combine them. When working with logarithms, we use algebraic strategies to simplify and solve mathematical expressions.
  • Algebra involves working with equations like \(2^x = 32\) and turning them into logarithmic form.
  • By using rules such as the Product Property, we transform multiplication into a simpler addition problem in logarithmic terms.
In the exercise, recognizing that \(\log_{2} (8 \cdot 4)\) and \(\log_{2} 32\) can be tackled using algebraic principles demonstrates this. Algebra, together with properties of logarithms, allows us to move fluidly between different forms of expression, making solutions clearer and more straightforward.

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Most popular questions from this chapter

The function \(P(x)=95-30 \log _{2} x\) models the percentage, \(P(x),\) of students who could recall the important features of a classroom lecture as a function of time, where \(x\) represents the number of days that have elapsed since the lecture was given. The figure at the top of the next column shows the graph of the function. Use this information to solve Exercises \(117-118\). Round answers to one decimal place. After how many days have all students forgotten the important features of the classroom lecture? (Let \(P(x)=0\) and solve for \(x\).) Locate the point on the graph that conveys this information.

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 your graphing utility's logarithmic regression option to obtain a model of the form \(y=a+b \ln x\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

Check each proposed solution by direct substitution or with a graphing utility. $$ (\log x)(2 \log x+1)=6 $$

Describe the change-of-base property and give an example.

Explaining the Concepts You take up weightlifting and record the maximum number of pounds you can lift at the end of each week. You start off with rapid growth in terms of the weight you can lift from week to week, but then the growth begins to level off. Describe how to obtain a function that models the number of pounds you can lift at the end of each week. How can you use this function to predict what might happen if you continue the sport?
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