Problem 150 - Algebra and Trigonometry

Question

Exercises 150–152 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 \(\log _{2}(8 \cdot 4) ?\)

Step-by-Step Solution

Answer

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

Step 1: 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, \(\log _{2} 32 = 5\).

Step 2: 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 \(3 + 2 = 5\)

Step 3: 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 \(\log _{2} 32 = \log _{2}(8 \cdot 4)\)

Key Concepts

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.