It is given that 6=3p^ and 8=3q^. Express log⁶288 in terms of p and q

Given that \( 6 = 3^p \) and \( 8 = 3^q \), we need to express \( \log_{6}(288) \) in terms of \( p \) and \( q \).

### Step 1: Factorize 288

First, let's factorize 288:

\[
288 = 2^5 \times 3^2
\]

### Step 2: Express \( \log_{6}(288) \) using the factorization

We can write:

\[
\log_{6}(288) = \log_{6}(2^5 \times 3^2)
\]

Using the logarithm property \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \):

\[
\log_{6}(288) = \log_{6}(2^5) + \log_{6}(3^2)
\]

Using the power rule \( \log_{b}(m^n) = n \log_{b}(m) \):

\[
\log_{6}(288) = 5 \log_{6}(2) + 2 \log_{6}(3)
\]

### Step 3: Express \( \log_{6}(2) \) and \( \log_{6}(3) \) in terms of \( p \) and \( q \)

Given \( 6 = 3^p \), we know:

\[
\log_{6}(3) = \frac{\log_{3}(3)}{\log_{3}(6)} = \frac{1}{p}
\]

Next, consider the expression for \( 8 = 3^q \), which leads to:

\[
\log_{6}(2) = \frac{\log_{3}(2)}{\log_{3}(6)} = \frac{q \cdot \log_{3}(2)}{\log_{3}(3^p)} = \frac{q \cdot \log_{3}(2)}{p \cdot \log_{3}(3)} = \frac{q \cdot \log_{3}(2)}{p}
\]

But since we don't have a direct expression for \( \log_{3}(2) \) in terms of \( p \) or \( q \), we'll need to rely on other relations.

### Step 4: Final expression for \( \log_{6}(288) \)

The final expression for \( \log_{6}(288) \) becomes:

\[
\log_{6}(288) = 5 \cdot \frac{\log_{6}(2)}{p} + 2 \cdot \frac{1}{p}
\]

Or more simplistically:

\[
\log_{6}(288) = \frac{5q}{p} \cdot \frac{\log_{3}(2)}{\log_{3}(3)} + \frac{2}{p}
\]

This result relies heavily on understanding the expressions \( \log_{6}(2) \) and \( \log_{6}(3) \) in terms of \( p \) and \( q \).

Would you like to explore this in greater detail?

### Follow-up Questions:
1. How can the properties of logarithms be used to express different bases?
2. What happens to logarithmic expressions when the base and argument are related by an exponential relationship?
3. How can logarithms be used to solve equations involving exponents?
4. Can the expression \( \log_{6}(288) \) be simplified further if additional information about \( p \) and \( q \) is known?
5. How would this problem change if the bases were irrational or complex?

### Tip:
When dealing with logarithms, it's often useful to express everything in terms of a common base if possible. This can make simplifications much easier and help in solving equations.

Learn how to express log base 6 of 288 in terms of p and q with a step-by-step solution. Understand logarithm properties and the power rule applied to logarithms. Suitable for advanced high school or college students studying logarithms and exponential functions.

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