if a>0,c>0,b=(ac)^1/2,a is not equal to 1,c is not equal to 1,ac is not equal to 1 and n>0,then the value of [ { (logn to the base a)-(logn to the base b) } / { (logn to the base b)-(logn to the base c) } ] is equal to:-A) (logn to the base a)/(log n to the base c) B) (loga to the base n)/(logc to the base n) C) (log a to the base c) D) none of these

To solve the expression given in your question, we need to manipulate logarithmic identities and apply some properties of logarithms. Let's break it down step by step to find the value of the expression:

Understanding the Expression

The expression we are working with is:

X =

We know that:

• b = (ac)^(1/2)
• log_n b = log_n (ac)^(1/2) = (1/2)(log_n a + log_n c)

Substituting for log_n b

Now, let's substitute log_n b into our expression:

X =

Simplifying the Numerator

For the numerator:

(log_n a) - (1/2)(log_n a + log_n c) = log_n a - (1/2)log_n a - (1/2)log_n c = (1/2)log_n a - (1/2)log_n c

Thus, the numerator simplifies to:

N = (1/2)(log_n a - log_n c)

Simplifying the Denominator

For the denominator:

(1/2)(log_n a + log_n c) - log_n c = (1/2)log_n a + (1/2)log_n c - log_n c = (1/2)log_n a - (1/2)log_n c

Thus, the denominator simplifies to:

D = (1/2)(log_n a - log_n c)

Final Expression

Now we can substitute N and D back into our expression for X:

X = \frac{(1/2)(log_n a - log_n c)}{(1/2)(log_n a - log_n c)} = 1

Analyzing the Options

Now, let's analyze the options provided:

• A) (log_n a)/(log_n c)
• B) (log_a n)/(log_c n)
• C) (log_a c)
• D) none of these

Since we simplified the expression to 1, and none of the options match this value, the correct answer is:

Answer: D) none of these

Conclusion

This problem illustrates the power of logarithmic properties in simplifying complex expressions. By carefully substituting and simplifying, we can arrive at a clear answer. If you have any further questions or need clarification on any steps, feel free to ask!