We are given the expressions $\log_7(8) = a$ and $\log_7(13) = b$. Using this information, we want to express $\log_{13}(8)$ in terms of $a$ and $b$.

Steps to solve:

1. Use the change-of-base formula: The change-of-base formula for logarithms states that: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$ where $k$ can be any base, but for our purposes, we will use base 7.

   So, we can rewrite $\log_{13}(8)$ as: $\log_{13}(8) = \frac{\log_7(8)}{\log_7(13)}$

2. Substitute known values: From the problem, we know: $\log_7(8) = a \quad \text{and} \quad \log_7(13) = b$ Substituting these into the equation: $\log_{13}(8) = \frac{a}{b}$

Thus, the expression for $\log_{13}(8)$ in terms of $a$ and $b$ is: $\log_{13}(8) = \frac{a}{b}$

Would you like more details or have any questions?

Related questions:

1. What is the change-of-base formula and why is it useful?
2. How would the solution differ if a different base were used for the logarithms?
3. How can logarithmic properties help in simplifying expressions like $\log_{13}(8)$?
4. Can you use this method for other log bases, such as natural logs or base 10 logs?
5. How does the property $\log_b(x^y) = y \log_b(x)$ factor into solving logarithmic equations?

Tip:

When dealing with logarithms of different bases, the change-of-base formula is a powerful tool for transforming and simplifying expressions.