To evaluate $\log_8 22$ using the change of base formula, we follow these steps.

Change of Base Formula:

The change of base formula for logarithms states that:

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

Where:

• $a$ is the number you are taking the logarithm of,
• $b$ is the base,
• $c$ is any base you choose (typically 10 or $e$).

For this case, we will use base 10 (common logarithm) to evaluate $\log_8 22$.

$\log_8 22 = \frac{\log_{10} 22}{\log_{10} 8}$

Now let's evaluate both $\log_{10} 22$ and $\log_{10} 8$ using a calculator.

1. $\log_{10} 22 \approx 1.3424$
2. $\log_{10} 8 \approx 0.9031$

Now substitute these values into the change of base formula:

$\log_8 22 = \frac{1.3424}{0.9031} \approx 1.486$

Final Answer:

$\log_8 22 \approx 1.486$

Would you like further details on any part of the solution?

Here are 5 related questions:

1. How would you evaluate $\log_5 50$ using the change of base formula?
2. Can the change of base formula be applied to any logarithmic base?
3. What is the value of $\log_8 64$?
4. How can we express $\log_8 22$ in terms of natural logarithms?
5. Why is the change of base formula useful in logarithmic calculations?

Tip: The change of base formula is especially helpful when calculating logarithms with uncommon bases, as most calculators only provide logarithms for base 10 and base $e$.