`loga/(y-z)=logb/(z-x)=logc/(x-y)` then value of `abc=`

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To solve the equation \( \frac{\log a}{y - z} = \frac{\log b}{z - x} = \frac{\log c}{x - y} \), we can follow these steps:

### Step 1: Set the common ratio
Let us denote the common ratio as \( \lambda \). Therefore, we can write:
\[ \log a = \lambda (y - z) \]
\[ \log b = \lambda (z - x) \]
\[ \log c = \lambda (x - y) \]

**Hint:** Assign a variable to the common ratio to simplify the equations.

### Step 2: Add the logarithmic equations
Now, we will add the three logarithmic equations together:
\[ \log a + \log b + \log c = \lambda (y - z) + \lambda (z - x) + \lambda (x - y) \]

**Hint:** Remember that the sum of logarithms can be combined into the logarithm of a product.

### Step 3: Simplify the right-hand side
The right-hand side simplifies as follows:
\[ \lambda (y - z + z - x + x - y) = \lambda (0) = 0 \]

**Hint:** Notice that the terms in the parentheses cancel each other out.

### Step 4: Conclude the logarithmic sum
Thus, we have:
\[ \log a + \log b + \log c = 0 \]

**Hint:** This indicates that the product of \( a, b, c \) is equal to 1.

### Step 5: Use properties of logarithms
Using the property of logarithms that states \( \log (abc) = 0 \) implies:
\[ abc = 10^0 = 1 \]

**Hint:** Recall that any number raised to the power of 0 is equal to 1.

### Final Result
The value of \( abc \) is:
\[ \boxed{1} \]