Problem 147 - College Algebra Essentials

Question

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. 5, b. 5, c. The two expressions are equivalent and prove the product rule of logarithms.

Step 1: Solving part a

We can write \(32\) as a power of \(2\) which is \(2^5\). So, we can replace \(32\) with \(2^5\) in our equation, thus, \(\log_2(2^5) = 5\) can be concluded using the fundamental property of Logarithms where \(\log_b(b^x) = x.\)

Step 2: Solving part b

We can simplify each term separately. We replace \(8\) and \(4\) with \(2^3\) and \(2^2\) respectively. The logarithmic expression becomes \(\log_2(2^3) + \log_2(2^2)\). From the fundamental logarithm property, we already know \(\log_2(2^2) = 2\) and \(\log_2(2^3) = 3\). Hence, the sum of these comes to \(2 + 3 =5\).

Step 3: Concluding part c

The expressions in part a and part b are equivalent, as \(\log_2 32 = \log_2(8 \cdot 4)\) and equals to \(5\). This serves as an illustration of the product rule of logarithms, which states that the log of the product of two numbers is equal to the sum of the logs of these two numbers.