Question

Evaluate log 3 square root 100

Answer 1

To evaluate \( \log_3(\sqrt{100}) \), we can first simplify \( \sqrt{100} \):

\[
\sqrt{100} = 10
\]

Now we can rewrite the expression:

\[
\log_3(\sqrt{100}) = \log_3(10)
\]

Next, we can use the change of base formula for logarithms, which states:

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

for any positive \( k \). Using base 10 (common logarithm) for convenience, we have:

\[
\log_3(10) = \frac{\log_{10}(10)}{\log_{10}(3)}
\]

Since \( \log_{10}(10) = 1 \), this simplifies to:

\[
\log_3(10) = \frac{1}{\log_{10}(3)}
\]

To evaluate \( \log_{10}(3) \), we can use a calculator:

\[
\log_{10}(3) \approx 0.4771
\]

Thus, we have:

\[
\log_3(10) \approx \frac{1}{0.4771} \approx 2.095
\]

Therefore, the value of \( \log_3(\sqrt{100}) \) is approximately \( 2.095 \).