Problem 139 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\log _{2} 8}{\log _{2} 4}=\frac{8}{4}$$

Step-by-Step Solution

Answer

The statement is false. The correct expression should be \(\frac{\log _{2} 8}{\log _{2} 4} = 1.5\), not 2.

Step 1: Evaluate each part of the expression

The first step is to evaluate each part of the given expression. \(\log _{2} 8\) translates to '2 to what power equals 8', the answer is 3. Similarly, \(\log _{2} 4\) translates to '2 to what power equals 4', the answer is 2. Hence, the left side of our equation becomes \(\frac{3}{2} = 1.5\).

Step 2: Evaluate the expression on the right

The second step is to solve the expression on the right, which is a simple calculation. \( \frac{8}{4} = 2\).

Step 3: Compare the left and the right side of the equation

Now we compare the simplified left side (\(1.5\)) with the right side (\(2\)). Since they are not equal, the given statement is false.