If `log_(0.3)(x-1)ltlog_(0.09)(x-1)`, then x lies in the interval

Options

A. `(2,oo)`
B. `(1,2)`
C. `(-2,-1)`
D. None of these

Answer

To solve the inequality \( \log_{0.3}(x-1) < \log_{0.09}(x-1) \), we will follow these steps:

### Step 1: Rewrite the logarithm
We can express \( 0.09 \) as \( (0.3)^2 \). Therefore, we can rewrite the inequality as:
\[ \log_{0.3}(x-1) < \log_{(0.3)^2}(x-1) \]

### Step 2: Use the change of base formula
Using the change of base formula for logarithms, we know that:
\[ \log_{(0.3)^2}(x-1) = \frac{\log_{0.3}(x-1)}{2} \]
Thus, we can rewrite the inequality:
\[ \log_{0.3}(x-1) < \frac{1}{2} \log_{0.3}(x-1) \]

### Step 3: Isolate the logarithm
To isolate \( \log_{0.3}(x-1) \), we can multiply both sides of the inequality by \( 2 \):
\[ 2 \log_{0.3}(x-1) < \log_{0.3}(x-1) \]

### Step 4: Rearranging the inequality
Rearranging gives us:
\[ 2 \log_{0.3}(x-1) - \log_{0.3}(x-1) < 0 \]
This simplifies to:
\[ \log_{0.3}(x-1) < 0 \]

### Step 5: Solve the logarithmic inequality
The inequality \( \log_{0.3}(x-1) < 0 \) implies that:
\[ x-1 < 1 \]
This means:
\[ x < 2 \]

### Step 6: Determine the domain of the logarithm
Since \( \log_{0.3}(x-1) \) is defined only when \( x-1 > 0 \), we have:
\[ x-1 > 0 \implies x > 1 \]

### Step 7: Combine the results
Combining the two inequalities, we have:
\[ 1 < x < 2 \]

### Final Answer
Thus, the solution is that \( x \) lies in the interval \( (1, 2) \).

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