Given the logarithms $\log_{b}(2) = p$, $\log_{b}(3) = q$, and $\log_{b}(5) = r$, we need to express $\log_{b}(0.001)$ in terms of $p$, $q$, and $r$.

First, notice that $0.001$ can be written as a power of 10:

$0.001 = 10^{-3}$

Thus, we have:

$\log_{b}(0.001) = \log_{b}(10^{-3})$

Using the logarithmic property $\log_{b}(a^c) = c \cdot \log_{b}(a)$, we can simplify:

$\log_{b}(10^{-3}) = -3 \cdot \log_{b}(10)$

Next, express $\log_{b}(10)$ in terms of $p$, $q$, and $r$. Note that:

$10 = 2 \times 5$

So:

$\log_{b}(10) = \log_{b}(2 \times 5) = \log_{b}(2) + \log_{b}(5)$

Using the values given:

$\log_{b}(10) = p + r$

Finally, substitute this into the expression for $\log_{b}(0.001)$:

$\log_{b}(0.001) = -3 \cdot (p + r)$

Thus, the expression for $\log_{b}(0.001)$ in terms of $p$ and $r$ is:

$\log_{b}(0.001) = -3(p + r)$

Would you like to see more details on logarithmic properties or have any questions?

Here are some related questions:

1. How do you express $\log_{b}(10)$ in terms of $p$ and $r$?
2. What is the relationship between logarithms of products and sums of logarithms?
3. Can you express $\log_{b}(0.2)$ in terms of $p$ and $r$?
4. How do logarithmic properties apply when dealing with powers of numbers?
5. What is the significance of the base in logarithmic functions?

Tip: Remember that logarithmic identities, like the product rule $\log_b(xy) = \log_b(x) + \log_b(y)$, are powerful tools for simplifying complex logarithmic expressions.