If `y=log_(10)x+log_(10)x^(1/3)+log_(10)x^(1/9)+ . . . and (2+4+6+ . . . +2y)/(3+6+9+ . . . +3y)=4/log_(10) x` then find the value of x and y

Options

A. `x=10^5 , y=8`
B. `x=10^6 , y=9`
C. `x=10^6 , y=8`
D. `x=10^5 , y=9`

Answer

To solve the given problem, we will follow these steps:

### Step 1: Express \( y \) in terms of \( \log_{10} x \)

Given:
\[ y = \log_{10} x + \log_{10} x^{1/3} + \log_{10} x^{1/9} + \ldots \]

Using the property of logarithms, we can rewrite the terms:
\[ y = \log_{10} x + \frac{1}{3} \log_{10} x + \frac{1}{9} \log_{10} x + \ldots \]

Factoring out \( \log_{10} x \):
\[ y = \log_{10} x \left(1 + \frac{1}{3} + \frac{1}{9} + \ldots \right) \]

### Step 2: Identify the series as a geometric series

The series \( 1 + \frac{1}{3} + \frac{1}{9} + \ldots \) is a geometric series with:
- First term \( a = 1 \)
- Common ratio \( r = \frac{1}{3} \)

The sum of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \]

Thus, we have:
\[ y = \log_{10} x \cdot \frac{3}{2} \]

### Step 3: Simplify the equation

Now we can express \( y \) as:
\[ y = \frac{3}{2} \log_{10} x \]

### Step 4: Analyze the second part of the equation

The second part of the problem is:
\[ \frac{2 + 4 + 6 + \ldots + 2y}{3 + 6 + 9 + \ldots + 3y} = \frac{4}{\log_{10} x} \]

Calculating the sums in the numerator and denominator:
- The numerator \( 2 + 4 + 6 + \ldots + 2y \) can be expressed as:
\[ 2(1 + 2 + 3 + \ldots + y) = 2 \cdot \frac{y(y + 1)}{2} = y(y + 1) \]

- The denominator \( 3 + 6 + 9 + \ldots + 3y \) can be expressed as:
\[ 3(1 + 2 + 3 + \ldots + y) = 3 \cdot \frac{y(y + 1)}{2} = \frac{3y(y + 1)}{2} \]

### Step 5: Substitute back into the equation

Now substituting these into the equation:
\[ \frac{y(y + 1)}{\frac{3y(y + 1)}{2}} = \frac{4}{\log_{10} x} \]

This simplifies to:
\[ \frac{2}{3} = \frac{4}{\log_{10} x} \]

### Step 6: Solve for \( \log_{10} x \)

Cross-multiplying gives:
\[ 2 \log_{10} x = 12 \implies \log_{10} x = 6 \]

### Step 7: Solve for \( x \)

Using the property of logarithms:
\[ x = 10^6 \]

### Step 8: Substitute \( x \) back to find \( y \)

Now substituting \( x = 10^6 \) back into the equation for \( y \):
\[ y = \frac{3}{2} \log_{10} (10^6) = \frac{3}{2} \cdot 6 = 9 \]

### Final Values

Thus, the values are:
\[ x = 10^6, \quad y = 9 \]