If `log_(10)(10x)=2.7532` then `log_(10) (10000x)` is

Options

A. 4.7532
B. 5.7532
C. `3xx2.7532`
D. none of these

Answer

To solve the problem, we need to find the value of \( \log_{10}(10000x) \) given that \( \log_{10}(10x) = 2.7532 \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \log_{10}(10x) = 2.7532 \]

2. **Use the property of logarithms**:
We can use the property of logarithms that states:
\[ \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \]
Applying this to our equation:
\[ \log_{10}(10x) = \log_{10}(10) + \log_{10}(x) \]
Therefore, we can rewrite the equation as:
\[ \log_{10}(10) + \log_{10}(x) = 2.7532 \]

3. **Calculate \( \log_{10}(10) \)**:
Since \( \log_{10}(10) = 1 \), we have:
\[ 1 + \log_{10}(x) = 2.7532 \]

4. **Isolate \( \log_{10}(x) \)**:
Subtract 1 from both sides:
\[ \log_{10}(x) = 2.7532 - 1 = 1.7532 \]

5. **Now, find \( \log_{10}(10000x) \)**:
We can express \( 10000x \) as \( 10000 \times x \). Using the logarithm property again:
\[ \log_{10}(10000x) = \log_{10}(10000) + \log_{10}(x) \]

6. **Calculate \( \log_{10}(10000) \)**:
Since \( 10000 = 10^4 \), we have:
\[ \log_{10}(10000) = 4 \]

7. **Substitute the values**:
Now substituting back into the equation:
\[ \log_{10}(10000x) = 4 + \log_{10}(x) \]
Substituting \( \log_{10}(x) = 1.7532 \):
\[ \log_{10}(10000x) = 4 + 1.7532 = 5.7532 \]

### Final Answer:
\[ \log_{10}(10000x) = 5.7532 \]