Question

Evaluate log ^3sqr tree 100

Answer 1

To evaluate \( \log_{\sqrt[3]{2}} 100 \), we can use the change of base formula for logarithms:

\[
\log_{\sqrt[3]{2}} 100 = \frac{\log_{10} 100}{\log_{10} \sqrt[3]{2}}
\]

First, we'll compute each of these logarithms.

1. **Calculate \( \log_{10} 100 \)**:
\[
\log_{10} 100 = 2
\]
(because \( 100 = 10^2 \))

2. **Calculate \( \log_{10} \sqrt[3]{2} \)**:
We can rewrite \( \sqrt[3]{2} \) as \( 2^{1/3} \):
\[
\log_{10} \sqrt[3]{2} = \log_{10} (2^{1/3}) = \frac{1}{3} \log_{10} 2
\]

Now, substituting these values back into the change of base formula gives:
\[
\log_{\sqrt[3]{2}} 100 = \frac{\log_{10} 100}{\log_{10} \sqrt[3]{2}} = \frac{2}{\frac{1}{3} \log_{10} 2} = \frac{2 \cdot 3}{\log_{10} 2} = \frac{6}{\log_{10} 2}
\]

To finalize the calculation, we can approximate \( \log_{10} 2 \):
\[
\log_{10} 2 \approx 0.301
\]

Thus:
\[
\log_{\sqrt[3]{2}} 100 \approx \frac{6}{0.301} \approx 19.93
\]

So the final result is:
\[
\log_{\sqrt[3]{2}} 100 \approx 19.93
\]

You can also finalize by expressing it as:
\[
\log_{\sqrt[3]{2}} 100 \approx \frac{6}{\log_{10} 2}
\]

For practical purposes with a calculator, you might get a closer approximation but the essential expression derived is accurate.