Problem 139 - College Algebra Essentials

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 \(\frac{\log_{2} 8}{\log_{2} 4}=\frac{8}{4}\) is false. The true statement should be \(\frac{\log_{2} 8}{\log_{2} 4} = \frac{3}{2}\).

Step 1: Solve Logarithmic Expressions

It's important to remember that logarithms are the inverse operations of exponential functions. Thus, \(\log_{2} 8\) simplifies to 3 because 2 raised to the power of 3 is equal to 8, and \(\log_{2} 4\) simplifies to 2 because 2 raised to the power of 2 is equal to 4. Hence, the given equation becomes \(\frac{3}{2}\).

Step 2: Compare to Incorrect Part of the Equation

Now, emphasizing on the incorrect part of the original equation that is \(\frac{8}{4}\), which simplifies to 2. It can be observed that \(\frac{3}{2}\) does not equal to 2.

Step 3: Provide the Correct Statement

To provide a correct statement, replace \(\frac{8}{4}\) with \(\frac{3}{2}\). This gives us the true equation: \(\frac{\log_{2} 8}{\log_{2} 4}=\frac{3}{2}\).