If `a^4b^5=1` then the value of `log_a(a^5b^4)` equals

Options

A. `9//5`
B. 4
C. 5
D. `8//5`

Explanation

To solve the problem, we need to find the value of \( \log_a(a^5b^4) \) given that \( a^4b^5 = 1 \).

### Step-by-Step Solution:

1. **Given Equation**: Start with the equation provided.
\[ a^4b^5 = 1 \]

2. **Taking Logarithm**: Take the logarithm of both sides. We can choose any base for the logarithm; here we will use base \( a \).
\[ \log_a(a^4b^5) = \log_a(1) \]

3. **Using Logarithm Properties**: Apply the properties of logarithms. The logarithm of a product can be expressed as the sum of the logarithms:
\[ \log_a(a^4) + \log_a(b^5) = 0 \]

4. **Simplifying the Logarithms**: Use the power rule of logarithms, which states \( \log_a(m^n) = n \log_a(m) \):
\[ 4 \log_a(a) + 5 \log_a(b) = 0 \]
Since \( \log_a(a) = 1 \):
\[ 4 \cdot 1 + 5 \log_a(b) = 0 \]
This simplifies to:
\[ 4 + 5 \log_a(b) = 0 \]

5. **Solving for \( \log_a(b) \)**: Rearranging the equation gives:
\[ 5 \log_a(b) = -4 \]
Dividing both sides by 5:
\[ \log_a(b) = -\frac{4}{5} \]

6. **Finding \( \log_a(a^5b^4) \)**: Now we need to find \( \log_a(a^5b^4) \):
\[ \log_a(a^5b^4) = \log_a(a^5) + \log_a(b^4) \]

7. **Applying Logarithm Properties Again**: Using the power rule again:
\[ \log_a(a^5) + \log_a(b^4) = 5 \log_a(a) + 4 \log_a(b) \]
Substituting \( \log_a(a) = 1 \) and \( \log_a(b) = -\frac{4}{5} \):
\[ 5 \cdot 1 + 4 \left(-\frac{4}{5}\right) = 5 - \frac{16}{5} \]

8. **Simplifying the Expression**: Convert 5 to a fraction:
\[ 5 = \frac{25}{5} \]
Thus:
\[ \frac{25}{5} - \frac{16}{5} = \frac{25 - 16}{5} = \frac{9}{5} \]

9. **Final Result**: Therefore, the value of \( \log_a(a^5b^4) \) is:
\[ \frac{9}{5} \]

### Final Answer:
\[ \log_a(a^5b^4) = \frac{9}{5} \]