A circle has radius `log_(10)(a^(2))` and a circumference of `log_(10)(b^(4))`. Then the value of `log_(a)b` is equal to :

Options

A. `(1)/(4pi)`
B. `(1)/(pi)`
C. `2pi`
D. `pi`

Answer

To solve the problem, we need to find the value of \( \log_a b \) given that the radius of the circle is \( \log_{10}(a^2) \) and the circumference is \( \log_{10}(b^4) \).

### Step-by-step Solution:

1. **Write down the formula for circumference**:
The circumference \( C \) of a circle is given by the formula:
\[ C = 2 \pi r \]
where \( r \) is the radius.

2. **Substitute the values**:
Here, the radius \( r \) is given as \( \log_{10}(a^2) \) and the circumference \( C \) is \( \log_{10}(b^4) \). Therefore, we can write:
\[ \log_{10}(b^4) = 2 \pi \cdot \log_{10}(a^2) \]

3. **Simplify the equation**:
We can simplify \( \log_{10}(a^2) \) using the logarithmic identity \( \log_{10}(a^m) = m \cdot \log_{10}(a) \):
\[ \log_{10}(b^4) = 2 \pi \cdot 2 \cdot \log_{10}(a) = 4 \pi \cdot \log_{10}(a) \]

4. **Rearranging the equation**:
Now we can rearrange the equation to isolate the logarithmic terms:
\[ \frac{\log_{10}(b^4)}{\log_{10}(a^2)} = 2\pi \]
This can be rewritten as:
\[ \log_{10}(b^4) = 2\pi \cdot \log_{10}(a^2) \]

5. **Using the change of base formula**:
We can express the left-hand side using the change of base formula:
\[ \log_{10}(b^4) = 4 \cdot \log_{10}(b) \]
Thus, we have:
\[ 4 \cdot \log_{10}(b) = 4 \pi \cdot \log_{10}(a) \]

6. **Dividing both sides by 4**:
Dividing both sides by 4 gives:
\[ \log_{10}(b) = \pi \cdot \log_{10}(a) \]

7. **Expressing in terms of \( \log_a b \)**:
We can express \( \log_a b \) using the change of base formula:
\[ \log_a b = \frac{\log_{10}(b)}{\log_{10}(a)} \]
Substituting the value of \( \log_{10}(b) \):
\[ \log_a b = \frac{\pi \cdot \log_{10}(a)}{\log_{10}(a)} = \pi \]

### Final Answer:
Thus, the value of \( \log_a b \) is:
\[ \log_a b = \pi \]