Find the value of `log_(10)7sqrt13` if `log_(10)13=1.1139`:

Options

A. a) 0.15913
B. b) 0.5119
C. c) 1.15913
D. d) none of these

Answer

To find the value of \( \log_{10} \left( 7 \sqrt{13} \right) \) given that \( \log_{10} 13 = 1.1139 \), we can follow these steps:

### Step 1: Rewrite the expression
We can express \( 7 \sqrt{13} \) as \( 7 \cdot 13^{1/2} \). Therefore, we can write:
\[ \log_{10} (7 \sqrt{13}) = \log_{10} (7 \cdot 13^{1/2}) \]

### Step 2: Apply the logarithm property
Using the property of logarithms that states \( \log_b (xy) = \log_b x + \log_b y \), we can separate the logarithm:
\[ \log_{10} (7 \cdot 13^{1/2}) = \log_{10} 7 + \log_{10} (13^{1/2}) \]

### Step 3: Simplify the logarithm of the power
Using the property \( \log_b (x^n) = n \cdot \log_b x \), we can simplify \( \log_{10} (13^{1/2}) \):
\[ \log_{10} (13^{1/2}) = \frac{1}{2} \log_{10} 13 \]

### Step 4: Substitute the known value
Now we substitute the known value of \( \log_{10} 13 \):
\[ \log_{10} (13^{1/2}) = \frac{1}{2} \cdot 1.1139 = 0.55695 \]

### Step 5: Combine the results
Now we can express the logarithm of \( 7 \sqrt{13} \):
\[ \log_{10} (7 \sqrt{13}) = \log_{10} 7 + 0.55695 \]

### Step 6: Find \( \log_{10} 7 \)
Since we do not have the value of \( \log_{10} 7 \) directly, we can use a calculator or logarithm tables to find \( \log_{10} 7 \). For this example, let's assume \( \log_{10} 7 \approx 0.8451 \) (you can verify this using a calculator).

### Step 7: Final calculation
Now we add the two logarithmic values:
\[ \log_{10} (7 \sqrt{13}) = 0.8451 + 0.55695 = 1.40205 \]

Thus, the value of \( \log_{10} (7 \sqrt{13}) \) is approximately \( 1.40205 \).