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

Options

A. `(2,oo)`
B. `(-2,-1)`
C. `(1,2)`
D. none

Answer

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

### Step 1: Identify the domain
Since we are dealing with logarithms, the argument must be greater than zero. Therefore, we need:
\[ x - 1 > 0 \implies x > 1 \]

**Hint:** Always check the domain of logarithmic functions to ensure the arguments are valid.

### Step 2: Rewrite the logarithms with the same base
We know that \( 0.09 = (0.3)^2 \). Thus, we can rewrite the logarithm with base \( 0.09 \) as:
\[ \log_{0.09}(x-1) = \log_{(0.3)^2}(x-1) = \frac{1}{2} \log_{0.3}(x-1) \]

### Step 3: Substitute back into the inequality
Now we can substitute this back into our original inequality:
\[ \log_{0.3}(x-1) < \frac{1}{2} \log_{0.3}(x-1) \]

### Step 4: Eliminate the logarithm
Since the base \( 0.3 \) is less than 1, the inequality sign will flip when we multiply both sides by 2:
\[ 2 \log_{0.3}(x-1) > \log_{0.3}(x-1) \]

### Step 5: Rearranging the inequality
Now, we can rearrange the inequality:
\[ 2 \log_{0.3}(x-1) - \log_{0.3}(x-1) > 0 \]
This simplifies to:
\[ \log_{0.3}(x-1) > 0 \]

### Step 6: Solve the logarithmic inequality
The inequality \( \log_{0.3}(x-1) > 0 \) implies:
\[ x - 1 < 1 \implies x < 2 \]

### Step 7: Combine the results
Now we have two conditions:
1. \( x > 1 \)
2. \( x < 2 \)

Combining these, we find:
\[ 1 < x < 2 \]

### Final Result
Thus, the solution for \( x \) lies in the interval:
\[ (1, 2) \]

### Summary of Steps
1. Identify the domain: \( x > 1 \)
2. Rewrite logarithm: \( \log_{0.09}(x-1) = \frac{1}{2} \log_{0.3}(x-1) \)
3. Substitute and simplify the inequality.
4. Eliminate the logarithm and rearrange.
5. Solve the resulting inequality.
6. Combine the results to find the final interval.