- If `log(5c/a),log((3b)/(5c))`and `log(a/(3b))`are in AP, where a, b, c are in GP, then a, b, c are the lengths ofsides of(A) an isosceles triangle(B) an equilateral triangle(D) none of these(C) a scalene triangle

Options

A. an isosceles triangle
B. an equilateral triangle
C. a scalene triangle
D. none of these

Explanation

To solve the problem, we need to determine the type of triangle formed by the sides \( a \), \( b \), and \( c \) given that the logarithmic expressions are in arithmetic progression (AP) and that \( a \), \( b \), and \( c \) are in geometric progression (GP).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We have three logarithmic expressions:
\[ \log\left(\frac{5c}{a}\right), \quad \log\left(\frac{3b}{5c}\right), \quad \log\left(\frac{a}{3b}\right) \]
These are stated to be in AP.

2. **Using the Property of AP**:
For three terms \( x, y, z \) to be in AP, it must hold that:
\[ 2y = x + z \]
Applying this to our logarithmic terms:
\[ 2\log\left(\frac{3b}{5c}\right) = \log\left(\frac{5c}{a}\right) + \log\left(\frac{a}{3b}\right) \]

3. **Using Logarithmic Properties**:
We can combine the logs on the right-hand side:
\[ \log\left(\frac{5c}{a}\right) + \log\left(\frac{a}{3b}\right) = \log\left(\frac{5c \cdot a}{a \cdot 3b}\right) = \log\left(\frac{5c}{3b}\right) \]
Thus, we can rewrite our equation as:
\[ 2\log\left(\frac{3b}{5c}\right) = \log\left(\frac{5c}{3b}\right) \]

4. **Exponentiating Both Sides**:
By exponentiating, we can eliminate the logarithm:
\[ \left(\frac{3b}{5c}\right)^2 = \frac{5c}{3b} \]

5. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 9b^2 = 25c^2 \]
This can be rearranged to:
\[ \frac{b^2}{c^2} = \frac{25}{9} \quad \Rightarrow \quad \frac{b}{c} = \frac{5}{3} \]

6. **Using the GP Condition**:
Since \( a, b, c \) are in GP, we can express \( b \) and \( c \) in terms of \( a \):
Let \( b = ar \) and \( c = ar^2 \) where \( r \) is the common ratio.
From the previous step, we have:
\[ \frac{ar}{ar^2} = \frac{5}{3} \quad \Rightarrow \quad \frac{1}{r} = \frac{5}{3} \quad \Rightarrow \quad r = \frac{3}{5} \]

7. **Finding the Sides**:
Now substituting \( r \) back:
\[ b = a \cdot \frac{3}{5}, \quad c = a \cdot \left(\frac{3}{5}\right)^2 = a \cdot \frac{9}{25} \]
Thus, the sides of the triangle are:
\[ a, \quad \frac{3a}{5}, \quad \frac{9a}{25} \]

8. **Determining the Type of Triangle**:
To determine if the triangle is scalene, isosceles, or equilateral, we check if all sides are different:
- \( a \) is the largest side.
- \( \frac{3a}{5} \) is less than \( a \).
- \( \frac{9a}{25} \) is less than \( \frac{3a}{5} \).

Since all three sides are different, the triangle is a **scalene triangle**.

### Conclusion:
The sides \( a \), \( \frac{3a}{5} \), and \( \frac{9a}{25} \) form a scalene triangle.