Let $$ \log_{b}\left(2\right)=p $$$$ \log_{b}\left(3\right)=q $$$$ \log_{b}\left(5\right)=r $$. Express $$ \log_5\left(b\right) $$in terms of p, q, or r

To express $\log_5(b)$ in terms of $p$, $q$, and $r$, we can use the change of base formula, which states:

\[
\log_a(c) = \frac{\log_b(c)}{\log_b(a)}
\]

Here, we want to express $\log_5(b)$. Using the change of base formula:

\[
\log_5(b) = \frac{\log_b(b)}{\log_b(5)}
\]

We know that $\log_b(b) = 1$ and $\log_b(5) = r$, so we have:

\[
\log_5(b) = \frac{1}{r}
\]

Thus, the expression for $\log_5(b)$ in terms of $r$ is:

\[
\log_5(b) = \frac{1}{r}
\]

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How can $\log_5(2)$ be expressed in terms of $p$ and $r$?
2. What is the relationship between $\log_b(30)$ and $p$, $q$, and $r$?
3. How would you express $\log_3(b)$ in terms of $p$, $q$, and $r$?
4. Can you express $\log_2(3)$ in terms of $p$ and $q$?
5. How do you find $\log_{b^2}(2)$ using $p$ and the change of base formula?

**Tip:** When solving logarithmic problems, the change of base formula is a powerful tool for converting logs to different bases, allowing for easier manipulation and comparison of logarithmic expressions.

Learn how to express $\log_5(b)$ in terms of given logarithmic values $p$, $q$, and $r$. Detailed step-by-step solution using the change of base formula. Suitable for high school students studying logarithms.

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