We are given the following:

$p = \log_b(3), \quad q = \log_b(6), \quad r = \log_b(10)$

We are asked to express the following logarithmic expressions in terms of $p$, $q$, and $r$.

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1. $\log_b(972)$

Start by factoring $972$:

$972 = 2^2 \times 3^5$

Now use the logarithmic property:

$\log_b(972) = \log_b(2^2 \times 3^5) = \log_b(2^2) + \log_b(3^5)$

Apply the power rule $\log_b(a^n) = n \log_b(a)$:

$\log_b(972) = 2 \log_b(2) + 5 \log_b(3)$

Next, express $\log_b(2)$ and $\log_b(3)$ in terms of $p$, $q$, and $r$. We know that $q = \log_b(6) = \log_b(2 \times 3) = \log_b(2) + \log_b(3)$, so:

$\log_b(2) = q - p$

Thus:

$\log_b(972) = 2(q - p) + 5p = 2q - 2p + 5p = 2q + 3p$

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2. $\log_b(27000b^2)$

Start by factoring $27000$:

$27000 = 2^3 \times 3^3 \times 5^3$

Now use the logarithmic property:

$\log_b(27000b^2) = \log_b(2^3 \times 3^3 \times 5^3 \times b^2)$

This can be rewritten as:

$\log_b(27000b^2) = \log_b(2^3) + \log_b(3^3) + \log_b(5^3) + \log_b(b^2)$

Apply the power rule:

$\log_b(27000b^2) = 3\log_b(2) + 3\log_b(3) + 3\log_b(5) + 2\log_b(b)$

We know $\log_b(b) = 1$, so this simplifies to:

$\log_b(27000b^2) = 3\log_b(2) + 3\log_b(3) + 3\log_b(5) + 2$

Next, express $\log_b(2)$, $\log_b(3)$, and $\log_b(5)$ in terms of $p$, $q$, and $r$. We know $\log_b(5) = r - p$, and $\log_b(2) = q - p$, so:

$\log_b(27000b^2) = 3(q - p) + 3p + 3(r - p) + 2$

Simplify:

$\log_b(27000b^2) = 3q - 3p + 3p + 3r - 3p + 2 = 3q + 3r - 3p + 2$

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3. $\log_b(0.02)$

We know that $0.02 = \frac{1}{50} = \frac{1}{2 \times 5^2}$, so:

$\log_b(0.02) = \log_b\left( \frac{1}{2 \times 5^2} \right)$

Using the logarithmic rule $\log_b\left( \frac{1}{x} \right) = -\log_b(x)$:

$\log_b(0.02) = -\log_b(2 \times 5^2) = -(\log_b(2) + \log_b(5^2))$

Apply the power rule:

$\log_b(0.02) = -(\log_b(2) + 2\log_b(5))$

Now express $\log_b(2)$ and $\log_b(5)$ in terms of $p$, $q$, and $r$:

$\log_b(0.02) = -(q - p + 2(r - p)) = -(q - p + 2r - 2p) = -(q + 2r - 3p)$

So:

$\log_b(0.02) = -q - 2r + 3p$

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Summary of Results:

1. $\log_b(972) = 2q + 3p$
2. $\log_b(27000b^2) = 3q + 3r - 3p + 2$
3. $\log_b(0.02) = -q - 2r + 3p$

Do you want further clarification or details on any of these steps?

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Related Questions:

1. How do logarithmic identities help in simplifying logarithmic expressions?
2. Can you express $\log_b(54)$ in terms of $p$, $q$, and $r$?
3. What are the properties of logarithms, and how can they be applied to solve complex expressions?
4. How does the change of base formula affect logarithmic expressions?
5. Can you calculate $\log_b(20)$ using the known values of $p$, $q$, and $r$?

Tip: When working with logarithms, always look for opportunities to apply properties like the product, quotient, and power rules to simplify the expressions.