To solve the problem $\log_8 \frac{1}{3}$ using the change of base formula, follow these steps:

The change of base formula is:

$\log_b a = \frac{\log a}{\log b}$

Here:

• $b = 8$
• $a = \frac{1}{3}$

Step 1: Apply the formula

$\log_8 \frac{1}{3} = \frac{\log \frac{1}{3}}{\log 8}$

Step 2: Compute $\log \frac{1}{3}$

Using the property of logarithms: $\log \frac{1}{3} = \log 1 - \log 3 = 0 - \log 3 = -\log 3$

Thus: $\log_8 \frac{1}{3} = \frac{-\log 3}{\log 8}$

Step 3: Compute $\log 3$ and $\log 8$

Using a calculator:

• $\log 3 \approx 0.4771$
• $\log 8 \approx 0.9031$

Substitute these values: $\log_8 \frac{1}{3} = \frac{-0.4771}{0.9031} \approx -0.5285$

Step 4: Round to the nearest thousandth

$\log_8 \frac{1}{3} \approx -0.529$

Final Answer:

$\boxed{-0.529}$

Let me know if you need further clarification or steps!

Expanding the topic:

1. How does the change of base formula work conceptually?
2. What is the property of logarithms used in simplifying $\log \frac{1}{3}$?
3. Can you apply this to $\log_2 0.25$?
4. Why are logarithmic functions important in real-world applications?
5. How do the properties of logarithms simplify calculations without a calculator?

Tip:

Always double-check your logarithmic values with a calculator or software for precision.