If x,y and z are in G.P. and `a^(x)=b^(y)=c^(z)` then

Options

A. `log_(b)a=log_(a)c`
B. `log_(c)b=log_(a)c`
C. `log_(b)a=log_(c)b`
D. none of these

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the Given Information
We are given that \( x, y, z \) are in geometric progression (G.P.) and that \( a^x = b^y = c^z \).

### Step 2: Take Logarithms
Taking logarithms on both sides of the equation \( a^x = b^y = c^z \):
\[ \log(a^x) = \log(b^y) = \log(c^z) \]

### Step 3: Apply Logarithm Properties
Using the property of logarithms, we can rewrite the equations:
\[ x \log a = y \log b = z \log c \]
Let’s denote this common value as \( k \):
\[ x \log a = k, \quad y \log b = k, \quad z \log c = k \]

### Step 4: Express \( x, y, z \) in terms of \( k \)
From the above equations, we can express \( x, y, z \) as:
\[ x = \frac{k}{\log a}, \quad y = \frac{k}{\log b}, \quad z = \frac{k}{\log c} \]

### Step 5: Use the Property of G.P.
Since \( x, y, z \) are in G.P., we have:
\[ y^2 = xz \]

### Step 6: Substitute the Values of \( x, y, z \)
Substituting the expressions for \( x, y, z \):
\[ \left(\frac{k}{\log b}\right)^2 = \left(\frac{k}{\log a}\right) \left(\frac{k}{\log c}\right) \]

### Step 7: Simplify the Equation
This leads to:
\[ \frac{k^2}{(\log b)^2} = \frac{k^2}{\log a \cdot \log c} \]

### Step 8: Cancel \( k^2 \) (Assuming \( k \neq 0 \))
Assuming \( k \neq 0 \), we can cancel \( k^2 \) from both sides:
\[ \frac{1}{(\log b)^2} = \frac{1}{\log a \cdot \log c} \]

### Step 9: Cross Multiply
Cross multiplying gives:
\[ \log a \cdot \log c = (\log b)^2 \]

### Step 10: Rewrite in Logarithmic Form
This can be rewritten using the property of logarithms:
\[ \frac{\log a}{\log b} = \frac{\log b}{\log c} \]

### Step 11: Conclude the Result
This implies:
\[ \log_a b = \log_b c \]

### Final Result
Thus, we conclude that:
\[ \log_a b = \log_b c \]

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