Problem 147 - College Algebra

Question

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

\(\log _{2} 32 = 5\), \(\log _{2} 8+ \log _{2} 4 = 5\), \(\log_2 32 = \log_2 8 + \log_2 4\). These expressions are equal, underlining the logarithmic rule \(\log_b (mn) = \log_b m + \log_b n\).

Step 1: Evaluate \(\log _{2} 32\)

Recall the rule that says \(\log_b b^y = y\). Knowing this, convert 32 to \(2^5\) and evaluate \(\log_2 2^5\), which simplifies to 5.

Step 2: Evalute \(\log _{2} 8+ \log _{2} 4\)

Using the same rule as above, convert 8 and 4 to values of the base 2. This gives \(\log_2 2^3 + \log_2 2^2\), which simplifies to 3 + 2, thus the total is 5.

Step 3: Draw a conclusion

The relationship between \(\log_2 32\) and \(\log_2 (8 \cdot 4) = \log_2 8 + \log_2 4\) is that they are equal. This demonstrates a property of logarithms: \(\log_b (mn) = \log_b m + \log_b n\). Here, \(m = 8, n = 4, b = 2\).