The value of x for which `log_(9)x - log_(9) (x/10 + 1/9)` is:

Options

A. 2
B. 4
C. 9
D. 10

Answer

To solve the equation \( \log_{9} x - \log_{9} \left( \frac{x}{10} + \frac{1}{9} \right) = 0 \), we can use the properties of logarithms.

### Step-by-Step Solution:

1. **Combine the Logarithms**:
Using the property of logarithms that states \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) \), we can rewrite the equation:
\[ \log_{9} \left( \frac{x}{\frac{x}{10} + \frac{1}{9}} \right) = 0 \]

2. **Exponentiate Both Sides**:
Since \( \log_{9} y = 0 \) implies \( y = 1 \), we can exponentiate both sides:
\[ \frac{x}{\frac{x}{10} + \frac{1}{9}} = 1 \]

3. **Cross Multiply**:
Cross multiplying gives us:
\[ x = \frac{x}{10} + \frac{1}{9} \]

4. **Eliminate the Fraction**:
To eliminate the fractions, multiply the entire equation by 90 (the least common multiple of 10 and 9):
\[ 90x = 9x + 10 \]

5. **Rearrange the Equation**:
Rearranging gives:
\[ 90x - 9x = 10 \]
\[ 81x = 10 \]

6. **Solve for x**:
Finally, divide both sides by 81:
\[ x = \frac{10}{81} \]

### Final Answer:
The value of \( x \) is \( \frac{10}{81} \).

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