If `(150)^(x) =7`, then x is equal to:

Options

A. `(log 7)/((log 3) + (log 5)+1)`
B. `(log 7)/((log 3) + (log 6))`
C. `(log 7)/((log 3) + (log 5) + 10)`
D. none of these

Answer

To solve the equation \( (150)^x = 7 \), we can follow these steps:

### Step-by-Step Solution:

1. **Take the logarithm of both sides**:
\[ \log((150)^x) = \log(7) \]

2. **Apply the power rule of logarithms**:
The power rule states that \( \log(a^b) = b \cdot \log(a) \). Applying this here gives:
\[ x \cdot \log(150) = \log(7) \]

3. **Solve for \( x \)**:
To isolate \( x \), divide both sides by \( \log(150) \):
\[ x = \frac{\log(7)}{\log(150)} \]

4. **Factor \( 150 \)**:
We can express \( 150 \) as \( 3 \times 5 \times 10 \):
\[ x = \frac{\log(7)}{\log(3 \times 5 \times 10)} \]

5. **Use the logarithm product rule**:
The product rule states that \( \log(a \times b) = \log(a) + \log(b) \). Applying this gives:
\[ x = \frac{\log(7)}{\log(3) + \log(5) + \log(10)} \]

6. **Recognize that \( \log(10) = 1 \)**:
Since we are using base 10 logarithms, we know that \( \log(10) = 1 \). Thus, we can simplify:
\[ x = \frac{\log(7)}{\log(3) + \log(5) + 1} \]

### Final Answer:
Thus, the value of \( x \) is:
\[ x = \frac{\log(7)}{\log(3) + \log(5) + 1} \]

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