The question asks you to evaluate the logarithmic expression:

$\log_{128} 32$

We need to evaluate the logarithm with base 128 of 32. Here's how we can do it:

Step 1: Express both numbers as powers of the same base

128 and 32 are both powers of 2. We can write them as:

• $128 = 2^7$
• $32 = 2^5$

Step 2: Rewrite the logarithmic expression

Now, we rewrite the logarithmic expression in terms of powers of 2: $\log_{128} 32 = \log_{2^7} 2^5$

Step 3: Use the change of base formula

The logarithm property $\log_b x = \frac{\log_k x}{\log_k b}$ allows us to change the base to 2: $\log_{2^7} 2^5 = \frac{\log_2 2^5}{\log_2 2^7}$

Step 4: Simplify using the power rule of logarithms

The power rule of logarithms $\log_b (a^n) = n \log_b a$ simplifies the expression: $\log_2 2^5 = 5 \quad \text{and} \quad \log_2 2^7 = 7$

Step 5: Final calculation

Thus, we get: $\log_{128} 32 = \frac{5}{7}$

So, the value of $\log_{128} 32$ is $\frac{5}{7}$.

Would you like more details on any of these steps? Or have any questions about logarithms in general?

Here are a few related questions for you to explore:

1. How would you evaluate $\log_2 64$?
2. What is the relationship between logarithms and exponents?
3. Can you solve logarithmic equations using the change of base formula?
4. What are some properties of logarithms that can simplify calculations?
5. How do you solve for an unknown in logarithmic form?

Tip: When you encounter logarithmic expressions with large bases, look for a way to express both numbers as powers of the same base to simplify the calculation!