If a `gt` 0, c `gt 0`, ` " b " = sqrt(ac)` and `" ac " ne 1`, ` N gt 0` , then `(log_(a) N - log_(b) N)/(log_(b) N - log_(c) N)` =

Options

A. `(log_(a) N)/(log_(c)N)`
B. `(log N)/(log C)`
C. `(log N )/(log a ) `
D. none of these

Answer

To solve the problem, we need to evaluate the expression:

\[ \frac{\log_a N - \log_b N}{\log_b N - \log_c N} \]

Given that \( b = \sqrt{ac} \) and \( ac \neq 1 \), we can use properties of logarithms to simplify the expression.

### Step 1: Rewrite the logarithms using change of base formula

Using the change of base formula, we can express the logarithms in terms of a common base (let's use \( N \)):

\[ \log_a N = \frac{1}{\log_N a}, \quad \log_b N = \frac{1}{\log_N b}, \quad \log_c N = \frac{1}{\log_N c} \]

### Step 2: Substitute the logarithms into the expression

Substituting these into our expression gives:

\[ \frac{\frac{1}{\log_N a} - \frac{1}{\log_N b}}{\frac{1}{\log_N b} - \frac{1}{\log_N c}} \]

### Step 3: Simplify the numerator and denominator

The numerator simplifies as follows:

\[ \frac{1}{\log_N a} - \frac{1}{\log_N b} = \frac{\log_N b - \log_N a}{\log_N a \cdot \log_N b} \]

The denominator simplifies similarly:

\[ \frac{1}{\log_N b} - \frac{1}{\log_N c} = \frac{\log_N c - \log_N b}{\log_N b \cdot \log_N c} \]

### Step 4: Combine the fractions

Now substituting these back into our expression gives:

\[ \frac{\frac{\log_N b - \log_N a}{\log_N a \cdot \log_N b}}{\frac{\log_N c - \log_N b}{\log_N b \cdot \log_N c}} \]

### Step 5: Simplify the overall expression

This simplifies to:

\[ \frac{(\log_N b - \log_N a) \cdot \log_N b \cdot \log_N c}{(\log_N c - \log_N b) \cdot \log_N a \cdot \log_N b} \]

Cancelling \( \log_N b \) from the numerator and denominator (since \( \log_N b \neq 0 \)) gives:

\[ \frac{\log_N b - \log_N a}{\log_N c - \log_N b} \]

### Step 6: Recognize the logarithmic identity

This can be rewritten using the property of logarithms:

\[ \frac{\log_N \frac{b}{a}}{\log_N \frac{c}{b}} \]

### Step 7: Final expression

Thus, we find that:

\[ \frac{\log_a N - \log_b N}{\log_b N - \log_c N} = \log_{\frac{c}{b}} \frac{b}{a} \]

### Conclusion

The final result is:

\[ \frac{\log_a N - \log_b N}{\log_b N - \log_c N} = 1 \]