Key Concepts and Formulas

1. Arithmetic Progression (A.P.) Property: If three numbers $X, Y, Z$ are in an A.P., then $2Y = X + Z$.
2. Logarithm Properties:
   • $\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$
   • $k \log_b M = \log_b (M^k)$
   • If $\log_b M = \log_b N$, then $M = N$.
3. Domain of Logarithms: For $\log_b X$ to be defined, $X > 0$.

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Step-by-Step Solution

Step 1: Interpret the first A.P. condition and derive a relationship between $a, b, c$.

We are given that $\log_e a, \log_e b, \log_e c$ are in an A.P. Using the A.P. property, twice the middle term is equal to the sum of the other two terms: $2 \log_e b = \log_e a + \log_e c$ Using the logarithm property $k \log_b M = \log_b (M^k)$ on the left side and $\log_b M + \log_b N = \log_b (MN)$ on the right side, we get: $\log_e (b^2) = \log_e (ac)$ Since the logarithms have the same base, their arguments must be equal: $b^2 = ac \quad (*)$ This implies that $a, b, c$ themselves form a Geometric Progression (G.P.).

Step 2: Interpret the second A.P. condition and simplify the terms.

We are given that $\log_e a - \log_e 2b$, $\log_e 2b - \log_e 3c$, $\log_e 3c - \log_e a$ are in an A.P. Let's simplify each term using the logarithm property $\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$: Term 1: $\log_e a - \log_e 2b = \log_e \left(\frac{a}{2b}\right)$ Term 2: $\log_e 2b - \log_e 3c = \log_e \left(\frac{2b}{3c}\right)$ Term 3: $\log_e 3c - \log_e a = \log_e \left(\frac{3c}{a}\right)$

Step 3: Apply the A.P. property to the simplified terms and derive a second relationship.

Since the three simplified terms are in an A.P., twice the middle term equals the sum of the first and the third term: $2 \left( \log_e \left(\frac{2b}{3c}\right) \right) = \log_e \left(\frac{a}{2b}\right) + \log_e \left(\frac{3c}{a}\right)$ Using the logarithm property $k \log_b M = \log_b (M^k)$ on the left side: $\log_e \left(\left(\frac{2b}{3c}\right)^2\right) = \log_e \left(\frac{4b^2}{9c^2}\right)$ Using the logarithm property $\log_b M + \log_b N = \log_b (MN)$ on the right side: $\log_e \left(\frac{a}{2b}\right) + \log_e \left(\frac{3c}{a}\right) = \log_e \left(\frac{a}{2b} \cdot \frac{3c}{a}\right) = \log_e \left(\frac{3c}{2b}\right)$ Equating the arguments of the logarithms: $\frac{4b^2}{9c^2} = \frac{3c}{2b}$

Step 4: Solve the derived equations to find the ratio $a:b:c$.

From Step 3, we have: $\frac{4b^2}{9c^2} = \frac{3c}{2b}$ Cross-multiply: $4b^2 \cdot 2b = 9c^2 \cdot 3c$ $8b^3 = 27c^3$ Taking the cube root of both sides: $2b = 3c$ This gives us a relationship between $b$ and $c$: $b = \frac{3}{2}c \quad \text{or} \quad c = \frac{2}{3}b$

Now, we use the relationship from Step 1, $b^2 = ac$, and substitute the expression for $c$: $b^2 = a \left(\frac{2}{3}b\right)$ Since $b$ must be positive for $\log_e b$ to be defined, we can divide both sides by $b$: $b = \frac{2}{3}a$ This gives us a relationship between $a$ and $b$: $a = \frac{3}{2}b$

Step 5: Express $a, b, c$ in terms of a common variable and find the ratio.

We have the following relationships: $a = \frac{3}{2}b$ $c = \frac{2}{3}b$

Let's express $a, b, c$ in terms of $b$: $a : b : c = \frac{3}{2}b : b : \frac{2}{3}b$ To get integer ratios, we can multiply all parts by the least common multiple of the denominators (2 and 3), which is 6: $a : b : c = \left(\frac{3}{2}b \cdot 6\right) : (b \cdot 6) : \left(\frac{2}{3}b \cdot 6\right)$ $a : b : c = (3b \cdot 3) : 6b : (2b \cdot 2)$ $a : b : c = 9b : 6b : 4b$ Dividing by $b$ (since $b > 0$), we get the ratio: $a : b : c = 9 : 6 : 4$

Step 6: Match the ratio with the given options.

The ratio $9:6:4$ corresponds to option (B).

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Common Mistakes & Tips

• Domain of Logarithms: Always remember that the arguments of logarithms must be positive. This implies $a, b, c > 0$.
• Algebraic Errors: Be careful with algebraic manipulations, especially when dealing with fractions and exponents.
• Misapplication of Logarithm Rules: Ensure you are using the correct logarithm properties for addition, subtraction, and multiplication by a constant.
• Checking the A.P. Condition: Double-check that you are correctly applying the $2Y = X + Z$ property to the terms of the A.P.

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Summary

The problem involves two conditions where terms are in an Arithmetic Progression. The first condition, involving logarithms of $a, b, c$, leads to the relationship $b^2 = ac$, indicating that $a, b, c$ form a Geometric Progression. The second condition, involving differences of logarithms, simplifies to a relationship between the arguments of these logarithms. By applying the A.P. property to these simplified logarithmic terms, we derive a second algebraic relationship between $a, b, c$. Solving these two relationships simultaneously allows us to determine the ratio $a:b:c$.

The final answer is \boxed{9: 6: 4}, which corresponds to option (B).