Question

14a) Use the change of base formula to complete the fraction.

logv5 37 =_/_

a. logv5 10
b. logv10 5
c. logv10 37
d. logv37 10

Evaluate logv5 37

A. 2.244
B. 1.568
C. 0.446
D. 0.699

Answer 1

The change of base formula for logarithms states that:

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

where \( k \) can be any positive number. In this case, we want to express \(\log_5 37\) using base 10 logarithms (common logarithm).

Thus, we can write:

\[
\log_5 37 = \frac{\log_{10} 37}{\log_{10} 5}
\]

So the correct answer for the fraction is:

\[
\log_5 37 = \frac{\log_{10} 37}{\log_{10} 5}
\]

Now we evaluate \(\log_5 37\). To find the numerical value of \(\log_5 37\), we can use a calculator to find \(\log_{10} 37\) and \(\log_{10} 5\):

\[
\log_{10} 37 \approx 1.568
\]
\[
\log_{10} 5 \approx 0.699
\]

Now we can calculate \(\log_5 37\):

\[
\log_5 37 \approx \frac{1.568}{0.699} \approx 2.244
\]

Thus, the evaluation of \(\log_5 37\) gives us approximately \(2.244\).

So the final answers are:

1. The fraction: \(\frac{\log_{10} 37}{\log_{10} 5}\)
2. The evaluated value of \(\log_5 37\): **A. 2.244**.