Key Concepts and Formulas

• Logarithm Properties:
  • Change of base: $\log_b a = \frac{\log_c a}{\log_c b}$
  • Power rule: $\log_b a^n = n \log_b a$
  • Definition: If $\log_b a = c$, then $a = b^c$.
• Arithmetic Progression (AP): A sequence of numbers such that the difference between the consecutive terms is constant. If $a_1, a_2, a_3, \dots$ are in AP with common difference $d$, then $a_n = a_1 + (n-1)d$.
• Exponential Equation Manipulation: Equating exponents when the bases are the same.

Step-by-Step Solution

Step 1: Utilize the Arithmetic Progression condition We are given that $3, 3\log_y x, 3\log_z y, 7\log_x z$ are in AP with a common difference $d = \frac{1}{2}$. The terms of the AP can be expressed as: First term: $a_1 = 3$ Second term: $a_2 = 3 + d = 3 + \frac{1}{2} = \frac{7}{2}$ Third term: $a_3 = 3 + 2d = 3 + 2(\frac{1}{2}) = 3 + 1 = 4$ Fourth term: $a_4 = 3 + 3d = 3 + 3(\frac{1}{2}) = 3 + \frac{3}{2} = \frac{9}{2}$

Now, we equate these terms with the given expressions: $3\log_y x = \frac{7}{2} \implies \log_y x = \frac{7}{6}$ (Equation 1) $3\log_z y = 4 \implies \log_z y = \frac{4}{3}$ (Equation 2) $7\log_x z = \frac{9}{2} \implies \log_x z = \frac{9}{14}$ (Equation 3)

Step 2: Express variables in terms of each other using logarithm definitions From the logarithm equations, we can convert them into exponential forms: From Equation 1: $x = y^{\frac{7}{6}}$ From Equation 2: $y = z^{\frac{4}{3}}$ From Equation 3: $z = x^{\frac{9}{14}}$

Step 3: Relate the given exponential equality using the derived logarithmic relationships We are given $x^{p q^{2}}=y^{q r}=z^{p^{2} r}$. Let's use the first part of the equality: $x^{p q^{2}}=y^{q r}$. Substitute $x = y^{\frac{7}{6}}$ into this equation: $(y^{\frac{7}{6}})^{p q^{2}} = y^{q r}$ $y^{\frac{7}{6} p q^{2}} = y^{q r}$ Equating the exponents, since the bases are the same: $\frac{7}{6} p q^{2} = q r$ Since q is a positive integer, we can divide by q: $\frac{7}{6} p q = r$ $7 p q = 6 r$ (Equation 4)

Now, let's use the second part of the equality: $y^{q r}=z^{p^{2} r}$. Substitute $y = z^{\frac{4}{3}}$ into this equation: $(z^{\frac{4}{3}})^{q r} = z^{p^{2} r}$ $z^{\frac{4}{3} q r} = z^{p^{2} r}$ Equating the exponents: $\frac{4}{3} q r = p^{2} r$ Since r is a positive integer, we can divide by r: $\frac{4}{3} q = p^{2}$ $4 q = 3 p^{2}$ (Equation 5)

Finally, let's use the relationship between x and z from the given equality: $x^{p q^{2}}=z^{p^{2} r}$. We know from Step 2 that $z = x^{\frac{9}{14}}$. Substitute this into the equality: $x^{p q^{2}} = (x^{\frac{9}{14}})^{p^{2} r}$ $x^{p q^{2}} = x^{\frac{9}{14} p^{2} r}$ Equating the exponents: $p q^{2} = \frac{9}{14} p^{2} r$ Since p is a positive integer, we can divide by p: $q^{2} = \frac{9}{14} p r$ $14 q^{2} = 9 p r$ (Equation 6)

Step 4: Solve the system of equations for p, q, and r We have the following equations:

1. $7 p q = 6 r$
2. $4 q = 3 p^{2}$
3. $14 q^{2} = 9 p r$
4. $r = p q + 1$ (Given in the problem statement)

Substitute Equation 4 into Equation 1: $7 p q = 6 (p q + 1)$ $7 p q = 6 p q + 6$ $p q = 6$ (Equation 7)

Now substitute Equation 7 into Equation 4 to find r: $r = 6 + 1 = 7$

From Equation 5, we have $q = \frac{3p^2}{4}$. Substitute this into Equation 7: $p \left(\frac{3p^2}{4}\right) = 6$ $\frac{3p^3}{4} = 6$ $3p^3 = 24$ $p^3 = 8$ Since p is a positive integer, $p = 2$.

Now substitute the value of p back into Equation 7 to find q: $2q = 6$ $q = 3$.

We have found $p=2$, $q=3$, and $r=7$. Let's verify these values with Equation 6: $14 q^{2} = 9 p r$ $14 (3^2) = 9 (2) (7)$ $14 (9) = 126$ $126 = 126$ The values are consistent.

Step 5: Calculate the final required value We need to find $r - p - q$. $r - p - q = 7 - 2 - 3 = 2$.

Common Mistakes and Tips

• Algebraic Errors: Be meticulous with algebraic manipulations, especially when dealing with exponents and fractions. Double-check each step.
• Logarithm Properties: Ensure a solid understanding and correct application of logarithm rules. Misapplying them can lead to incorrect relationships between variables.
• Substitution Strategy: Carefully choose which equations to substitute into others to simplify the problem efficiently. Sometimes, a direct substitution might be more complex than an indirect one.
• Verification: After finding potential values for p, q, and r, it's a good practice to plug them back into the original conditions to ensure they satisfy all constraints.

Summary The problem requires a systematic approach involving the properties of arithmetic progressions and logarithms. By first extracting relationships between the logarithms of x, y, and z from the AP condition, we then used these relationships to simplify the given exponential equality. This led to a system of algebraic equations involving p, q, and r, which were then solved using the additional condition $r = pq + 1$. The derived values of p, q, and r were then used to compute the final answer.

The final answer is \boxed{2}.