What value is equivalent to log75.1

Question

log
7
⁡
5.1
to the nearest thousandth? Use the change of base formula to determine your answer.(1 point)
Responses

6.035
6.035

1.194
1.194

0.837
0.837

0.101

Answer 1

To find the value of \( \log_7(5.1) \) using the change of base formula, we can use:

\[ \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \]

where \( k \) can be any positive number (commonly, we use 10 or e). We will use base 10 for our calculation:

\[ \log_7(5.1) = \frac{\log_{10}(5.1)}{\log_{10}(7)} \]

Now let's calculate the logarithms:

1. Calculate \( \log_{10}(5.1) \):

Using a calculator: \[ \log_{10}(5.1) \approx 0.70757 \]

2. Calculate \( \log_{10}(7) \):

Using a calculator: \[ \log_{10}(7) \approx 0.84510 \]

Now, we substitute these values into the change of base formula:

\[ \log_7(5.1) \approx \frac{0.70757}{0.84510} \approx 0.837 \]

Thus, the value of \( \log_7(5.1) \) to the nearest thousandth is approximately:

\[ \boxed{0.837} \]