$$ \log_{b}\left(2\right)=p $$$$ \log_{b}\left(3\right)=q $$$$ \log_{b}\left(5\right)=r $$Express $$ \log_{b}\left(.001\right) $$in terms of p, q, or r.

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.

Learn how to express logarithms of decimals in terms of given values using logarithmic properties. This problem involves calculating $\log_{b}(0.001)$ based on $\log_{b}(2) = p$, $\log_{b}(3) = q$, and $\log_{b}(5) = r$. Suitable for high school students studying advanced algebra and logarithms.

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