Question

The number of non-negative values of n for which \(\log_{1/4}(n^2 - 7n + 14)>0\) is ____.

Hint:

For logarithmic inequalities with a base between 0 and 1, remember to reverse the inequality sign. Analyzing the minimum or maximum value of a quadratic argument is often the quickest way to solve.

Answer 1

Correct Answer: 0

Step 1: Understanding the Question 
We are tasked with solving a logarithmic inequality. The key features include:

• The base of the logarithm, which is between 0 and 1.
• The argument of the logarithm is a quadratic expression.

Step 2: Key Formula or Approach
For a logarithmic inequality \( \log_b(A) > C \), where the base \( b \) is between 0 and 1 (\( 0 < b < 1 \)), the inequality sign is reversed when converting to exponential form. Also, the argument \( A \) must be positive.
Therefore, \( \log_b(A) > C \) becomes: \[ 0 < A < b^C \] In our case, \( b = \frac{1}{4} \), \( A = n^2 - 7n + 14 \), and \( C = 0 \). Applying the rule, the inequality \( \log_{1/4}(n^2 - 7n + 14) > 0 \) becomes: \[ 0 < n^2 - 7n + 14 < \left(\frac{1}{4}\right)^0 \] \[ 0 < n^2 - 7n + 14 < 1 \] This results in two separate inequalities to solve:

• 1) \( n^2 - 7n + 14 > 0 \)
• 2) \( n^2 - 7n + 14 < 1 \) or \( n^2 - 7n + 13 < 0 \)

Step 3: Detailed Explanation
Let's analyze the quadratic expression \( f(n) = n^2 - 7n + 14 \). This is an upward-opening parabola. We can find its minimum value by finding the vertex.
The vertex occurs at: \[ n = -\frac{-7}{2(1)} = 3.5 \] The minimum value of the expression is: \[ f(3.5) = (3.5)^2 - 7(3.5) + 14 = 12.25 - 24.5 + 14 = 1.75 \] Since the minimum value of \( n^2 - 7n + 14 \) is 1.75, the expression is always greater than 0. Therefore, the first inequality \( n^2 - 7n + 14 > 0 \) is always true.

Now, for the second inequality \( n^2 - 7n + 14 < 1 \), since the minimum value of the left side is 1.75, it can never be less than 1. Thus, the second inequality has no solution.

Since there are no real values of \( n \) that satisfy the system, there are no non-negative values either.

Step 4: Final Answer
The number of non-negative values of \( n \) is 0.