Key Concepts and Formulas

• Logarithm Properties: $\log_b (M^p) = p \log_b M$.
• Infinite Geometric Series: The sum of an infinite geometric series with first term $a$ and common ratio $r$ (where $|r| < 1$) is $S_\infty = \frac{a}{1-r}$.
• Sum of First n Natural Numbers: $1 + 2 + \dots + n = \frac{n(n+1)}{2}$.
• Arithmetic Progression (AP) Sum: The sum of an AP can be found by factoring out a common term and using the sum of natural numbers formula.

Step-by-Step Solution

Step 1: Simplify the expression for $y$ The given equation for $y$ is $y = \log_{10} x + \log_{10} x^{1/3} + \log_{10} x^{1/9} + \dots$ Using the logarithm property $\log_b (M^p) = p \log_b M$, we can rewrite each term: $y = \log_{10} x + \frac{1}{3}\log_{10} x + \frac{1}{9}\log_{10} x + \dots$ We can factor out $\log_{10} x$: $y = (\log_{10} x) \left(1 + \frac{1}{3} + \frac{1}{9} + \dots \right)$ The series in the parenthesis is an infinite geometric progression with first term $a=1$ and common ratio $r = \frac{1}{3}$. Since $|r| < 1$, the sum to infinity is: $S_\infty = \frac{a}{1-r} = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2}$ Substituting this back into the equation for $y$: $y = (\log_{10} x) \times \frac{3}{2}$ This gives us our first important relation: $\log_{10} x = \frac{2y}{3}$ (Equation 1)

Step 2: Simplify the left side of the second equation The left side of the second equation is $\frac{2 + 4 + 6 + \dots + 2y}{3 + 6 + 9 + \dots + 3y}$. The numerator is $2 + 4 + 6 + \dots + 2y$. We can factor out 2: Numerator = $2(1 + 2 + 3 + \dots + y)$ Using the formula for the sum of the first $y$ natural numbers, $1 + 2 + \dots + y = \frac{y(y+1)}{2}$: Numerator = $2 \times \frac{y(y+1)}{2} = y(y+1)$

The denominator is $3 + 6 + 9 + \dots + 3y$. We can factor out 3: Denominator = $3(1 + 2 + 3 + \dots + y)$ Using the same formula for the sum of the first $y$ natural numbers: Denominator = $3 \times \frac{y(y+1)}{2}$

Now, substitute these simplified expressions back into the fraction: $\frac{\text{Numerator}}{\text{Denominator}} = \frac{y(y+1)}{3 \times \frac{y(y+1)}{2}}$ Assuming $y > 0$, we can cancel out $y(y+1)$: $\frac{1}{3/2} = \frac{2}{3}$

Step 3: Solve the second equation The second equation is $\frac{2 + 4 + 6 + \dots + 2y}{3 + 6 + 9 + \dots + 3y} = \frac{4}{\log_{10} x}$. From Step 2, we found the left side simplifies to $\frac{2}{3}$. So, we have: $\frac{2}{3} = \frac{4}{\log_{10} x}$ Now, we solve for $\log_{10} x$: $2 \times \log_{10} x = 3 \times 4$ $2 \log_{10} x = 12$ $\log_{10} x = 6$ (Equation 2)

Step 4: Solve the system of equations for $x$ and $y$ We have two equations:

1. $\log_{10} x = \frac{2y}{3}$
2. $\log_{10} x = 6$

From Equation 2, we directly get $\log_{10} x = 6$. Substitute this value into Equation 1: $6 = \frac{2y}{3}$ Multiply both sides by 3: $18 = 2y$ Divide by 2: $y = 9$

Now we find $x$ using $\log_{10} x = 6$. By the definition of logarithms, this means: $x = 10^6$

Step 5: Form the ordered pair $(x, y)$ The ordered pair $(x, y)$ is $(10^6, 9)$.

Common Mistakes & Tips

• Logarithm Properties: Ensure accurate application of $\log_b (M^p) = p \log_b M$. Forgetting this will lead to incorrect series simplification.
• GP Identification: Carefully identify the first term ($a$) and common ratio ($r$) of the geometric progression. A common error is miscalculating $r$.
• AP Sum Simplification: Factoring out common terms from APs, like 2 from the numerator and 3 from the denominator, simplifies the calculation significantly and reduces the chance of errors.
• Solving the System: Treat the two derived equations as a system of equations and solve them systematically.

Summary The problem requires combining logarithmic properties with the sums of infinite geometric and arithmetic progressions. First, we simplified the expression for $y$ by recognizing an infinite geometric series, leading to a relationship between $y$ and $\log_{10} x$. Then, we simplified the ratio in the second equation by recognizing arithmetic progressions and using the formula for the sum of natural numbers. This yielded a direct value for $\log_{10} x$. Finally, we solved the system of two equations to find the values of $x$ and $y$.

The final answer is \boxed{\text{(10^6, 9)}}.