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The problem provided seems to involve properties of logarithms. Here's how to solve it using logarithmic rules:

**Problem 7:**
\[ 2\log(81) - 5\log(2) - 3\log(5) \]

**Step 1:** Apply the power rule of logarithms, which states \( a\log(b) = \log(b^a) \), to rewrite the expression:
\[ \log(81^2) - \log(2^5) - \log(5^3) \]

**Step 2:** Convert the numbers to their exponential forms:
\[ \log(81^2) = \log(6561), \quad 2^5 = 32, \quad 5^3 = 125 \]

**Step 3:** Continue with the expression:
\[ \log(6561) - \log(32) - \log(125) \]

**Step 4:** Use the quotient rule of logarithms, which states \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \), to combine the logs:
\[ \log\left(\frac{6561}{32 \times 125}\right) \]

**Step 5:** Compute the value inside the logarithm:
\[ \frac{6561}{32 \times 125} = \frac{6561}{4000} = 1.64025 \]

**Step 6:** Substitute back to complete the logarithmic form:
\[ \log(1.64025) \]

Therefore, the solution to problem 7 is \( \log(1.64025) \).

Would you like more details on this solution or help with any other question? Here are some follow-up questions and a tip:

1. How can the logarithmic properties be used to simplify expressions like \( \log(a) + \log(b) \)?
2. What would be the inverse operation of each logarithmic rule applied here?
3. How do logarithms apply in real-world situations or other mathematical fields?
4. What are some common mistakes to avoid when manipulating logarithmic expressions?
5. Can we explore logarithms in different bases, such as natural logarithms (ln) versus common logarithms (log)?

**Tip:** Always remember to simplify expressions inside the logarithm as much as possible before applying logarithmic operations to make calculations easier.

Solve the logarithmic expression: 2log 81 · 5log 2 · 3log 5

Learn how to solve the logarithmic expression 2log(81) · 5log(2) · 3log(5) using logarithmic rules, including the power rule and quotient rule. This problem is suitable for high school students, covering key logarithmic concepts and step-by-step solutions.

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