Problem 113 - Introductory and Intermediate Algebra for College Students 4th

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 statement is: \(\frac{\log_2{8}}{\log_2{4}} = \frac{3}{2}\)

Step 1: Simplify the Logarithms

Recall that \( \log_b{a^n} = n \cdot \log_b{a}\). Using this property for the numerator gives \(\log_2{8} = 3 \cdot \log_2{2}\) which simplifies to 3 since \(\log_b{b} = 1\). Similarly, for the denominator, \(\log_2{4} = 2 \cdot \log_2{2}\) simplifies to 2. So, the fraction simplifies to \(\frac{3}{2}\).

Step 2: Evaluate the Right Side of the Equation

Calculate the right side of the equation \( \frac{8}{4} = 2\)

Step 3: Compare the Results and Make Necessary Changes

Comparing the result from Step 1 (\(\frac{3}{2}\)) and Step 2 (2), it's obvious that the original statement is false. Therefore, the statement needed to be changed as follows to make it true: \(\frac{\log _{2} 8}{\log _{2} 4}=\frac{3}{2}\)