If `log_(0.04) (x-1)>=log_(0.2) (x-1)` then `x` belongs to the interval

Options

A. `(1, 2]`
B. `(-oo, 2]`
C. `[2, oo)`
D. none of these

Explanation

To solve the inequality \( \log_{0.04}(x-1) \geq \log_{0.2}(x-1) \), we will follow these steps:

### Step 1: Rewrite the logarithms in terms of a common base
We know that \( 0.04 = (0.2)^2 \). Therefore, we can rewrite the logarithm:

\[ \log_{0.04}(x-1) = \log_{(0.2)^2}(x-1) = \frac{1}{2} \log_{0.2}(x-1) \]

### Step 2: Substitute into the inequality
Now we can substitute this into our original inequality:

\[ \frac{1}{2} \log_{0.2}(x-1) \geq \log_{0.2}(x-1) \]

### Step 3: Eliminate the logarithm
To eliminate the logarithm, we can multiply both sides by 2 (since \( \log_{0.2}(x-1) \) is positive when \( x-1 > 1 \)):

\[ \log_{0.2}(x-1) \geq 0 \]

### Step 4: Solve the logarithmic inequality
The inequality \( \log_{0.2}(x-1) \geq 0 \) implies:

\[ x-1 \leq 1 \]

This is because the logarithm with a base less than 1 is decreasing. Therefore, we can rewrite this as:

\[ x \leq 2 \]

### Step 5: Determine the domain of the logarithm
Since we are dealing with logarithms, we must also consider the domain. The expression \( x-1 \) must be greater than 0:

\[ x-1 > 0 \implies x > 1 \]

### Step 6: Combine the results
Now we combine the two results:

1. \( x > 1 \)
2. \( x \leq 2 \)

Thus, the solution for \( x \) is:

\[ 1 < x \leq 2 \]

### Final Answer
The interval for \( x \) is \( (1, 2] \).

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