A line x=k intersects the graph of `y=log_4 x` and `y=log_4 (x + 4)`. The distance between the points of intersection is 0.5, then the value of k is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem step by step, we can follow these procedures:

### Step 1: Identify the Points of Intersection
The line \( x = k \) intersects the graphs of \( y = \log_4 x \) and \( y = \log_4 (x + 4) \) at two points:
- Point A: \( (k, \log_4 k) \)
- Point B: \( (k, \log_4 (k + 4)) \)

### Step 2: Calculate the Distance Between Points A and B
The distance \( d \) between points A and B can be calculated using the distance formula. Since both points have the same x-coordinate (k), the distance is given by the difference in their y-coordinates:

\[ d = |y_2 - y_1| = |\log_4 (k + 4) - \log_4 k| \]

### Step 3: Use the Logarithmic Property
Using the property of logarithms, we can simplify the expression for distance:

\[ d = |\log_4 (k + 4) - \log_4 k| = |\log_4 \left(\frac{k + 4}{k}\right)| \]

### Step 4: Set the Distance Equal to 0.5
According to the problem, the distance is given as 0.5:

\[ |\log_4 \left(\frac{k + 4}{k}\right)| = 0.5 \]

### Step 5: Remove the Absolute Value
Since logarithms can be positive or negative, we can write two equations:

1. \(\log_4 \left(\frac{k + 4}{k}\right) = 0.5\)
2. \(\log_4 \left(\frac{k + 4}{k}\right) = -0.5\)

### Step 6: Solve the First Equation
Starting with the first equation:

\[ \log_4 \left(\frac{k + 4}{k}\right) = 0.5 \]

This implies:

\[ \frac{k + 4}{k} = 4^{0.5} = 2 \]

Cross-multiplying gives:

\[ k + 4 = 2k \]

Rearranging leads to:

\[ k = 4 \]

### Step 7: Solve the Second Equation
Now, solving the second equation:

\[ \log_4 \left(\frac{k + 4}{k}\right) = -0.5 \]

This implies:

\[ \frac{k + 4}{k} = 4^{-0.5} = \frac{1}{2} \]

Cross-multiplying gives:

\[ k + 4 = \frac{k}{2} \]

Rearranging leads to:

\[ 2(k + 4) = k \implies 2k + 8 = k \implies k = -8 \]

### Step 8: Conclusion
The value of \( k \) must be positive since it represents a point on the graph of a logarithm. Therefore, the only valid solution is:

\[ \boxed{4} \]

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