We need to evaluate the logarithm $\log_8(16)$.

Step 1: Express 16 and 8 as powers of 2

• $8 = 2^3$
• $16 = 2^4$

Step 2: Rewrite the logarithmic expression using these powers of 2

We can now rewrite the logarithmic expression as:

$\log_8(16) = \log_{2^3}(2^4)$

Step 3: Apply the change of base formula

The change of base formula for logarithms is:

$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$

Using base 2:

$\log_{2^3}(2^4) = \frac{\log_2(2^4)}{\log_2(2^3)}$

Step 4: Simplify using properties of logarithms

• $\log_2(2^4) = 4$
• $\log_2(2^3) = 3$

Thus, we have:

$\log_{2^3}(2^4) = \frac{4}{3}$

Final Answer:

$\log_8(16) = \frac{4}{3}$

Would you like a detailed explanation of any of the steps, or have further questions?

Here are some related questions for exploration:

1. How do logarithmic functions relate to exponential functions?
2. What is the inverse of the logarithm function?
3. How do you solve logarithms with different bases?
4. Can you simplify logarithms with fractional exponents?
5. What are the properties of logarithms (product, quotient, and power rules)?

Tip: When working with logarithms, it's often helpful to rewrite the numbers involved as powers of the same base to simplify the calculation.