Questions: Evaluate the expression log base 8 of 2.

Solution Steps

To evaluate the expression \(\log_{8} 2\), we can use the change of base formula for logarithms, which states that \(\log_{a} b = \frac{\log_{c} b}{\log_{c} a}\). We can choose any base \(c\), but base 10 or base \(e\) (natural logarithm) are commonly used.

To evaluate the expression \(\log_{8} 2\), we use the change of base formula: \[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \] We choose the natural logarithm (base \(e\)) for convenience: \[ \log_{8} 2 = \frac{\ln 2}{\ln 8} \]

Next, we calculate the natural logarithms: \[ \ln 2 \approx 0.6931 \] \[ \ln 8 \approx 2.0794 \]

Now, we divide the natural logarithm of 2 by the natural logarithm of 8: \[ \log_{8} 2 = \frac{\ln 2}{\ln 8} \approx \frac{0.6931}{2.0794} \approx 0.3333 \]

Final Answer