Suppose that 𝑝=log𝑏⁡(3), 𝑞=log𝑏⁡(6), and 𝑟=log𝑏⁡(10). Express the following logarithmic expressions in terms of 𝑝, 𝑞, and 𝑟. logb(972), logb(27000b^2), logb(.02)

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:

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

---

### 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:

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

---

### 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:

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

---

### 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.

Suppose that p = log_b(3), q = log_b(6), and r = log_b(10). Express the following logarithmic expressions in terms of p, q, and r: log_b(972), log_b(27000b^2), log_b(0.02).

Explore how to express complex logarithmic expressions in terms of simpler variables (p, q, and r) using logarithmic identities. This solution involves factoring numbers, applying product, power, and quotient rules, and simplifying logarithmic expressions like log_b(972), log_b(27000b^2), and log_b(0.02). Suitable for high school students in Grades 10-12.

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