The value of `a^((log_b(log_bx))/(log_b a)),` is

Options

A. `log_aN`
B. `log_bN`
C. `log_Na`
D. `log_Nb`

Answer

To find the value of \( a^{\frac{\log_b(\log_b x)}{\log_b a}} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ a^{\frac{\log_b(\log_b x)}{\log_b a}} \]

### Step 2: Use the change of base formula
According to the change of base formula for logarithms, we can express \(\log_b a\) as:
\[ \log_b a = \frac{\log_k a}{\log_k b} \]
for any base \(k\). Here, we will use base \(k = b\) for simplicity. Thus, we can rewrite \(\log_b a\) as:
\[ \log_b a = \frac{\log_b a}{\log_b b} = \log_b a \]
This means we can keep \(\log_b a\) as is.

### Step 3: Rewrite the exponent
Now, we can rewrite the exponent:
\[ \frac{\log_b(\log_b x)}{\log_b a} = \log_a(\log_b x) \]
This follows from the property of logarithms where \(\frac{\log_b x}{\log_b a} = \log_a x\).

### Step 4: Substitute back into the expression
Substituting this back into our expression gives us:
\[ a^{\log_a(\log_b x)} \]

### Step 5: Apply the property of exponents
Using the property \(a^{\log_a x} = x\), we can simplify:
\[ a^{\log_a(\log_b x)} = \log_b x \]

### Final Result
Thus, the value of \( a^{\frac{\log_b(\log_b x)}{\log_b a}} \) is:
\[ \log_b x \]

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