To simplify the expression \( 2\log{a} + 5\log{b} - \log{c} \), we can use the following logarithmic properties:

1. **Power Rule**: \( n\log{x} = \log{x^n} \)
2. **Product Rule**: \( \log{x} + \log{y} = \log{xy} \)
3. **Quotient Rule**: \( \log{x} - \log{y} = \log{\frac{x}{y}} \)

### Step-by-step solution:

1. Apply the **Power Rule**:
   \[
   2\log{a} = \log{a^2}
   \]
   \[
   5\log{b} = \log{b^5}
   \]

   So, the expression becomes:
   \[
   \log{a^2} + \log{b^5} - \log{c}
   \]

2. Apply the **Product Rule** to combine \( \log{a^2} + \log{b^5} \):
   \[
   \log{a^2} + \log{b^5} = \log{(a^2 \cdot b^5)}
   \]

   Now, the expression is:
   \[
   \log{(a^2 \cdot b^5)} - \log{c}
   \]

3. Apply the **Quotient Rule**:
   \[
   \log{(a^2 \cdot b^5)} - \log{c} = \log{\left( \frac{a^2 \cdot b^5}{c} \right)}
   \]

### Final simplified expression:
\[
\log{\left( \frac{a^2 \cdot b^5}{c} \right)}
\]

Would you like any further details or have any questions?

Here are 5 related questions to further explore the topic:

1. How do you solve equations involving logarithms?
2. What are the conditions for logarithms to be valid?
3. How do you convert logarithmic equations to exponential form?
4. What is the natural logarithm, and how is it different from base 10 logarithms?
5. How do you apply logarithms in real-world problems like population growth?

**Tip:** Always check the domain of a logarithmic expression to ensure that the arguments inside the log function are positive.

Simplify the logarithmic expression 2log(a) + 5log(b) - log(c) using key logarithmic properties such as the Power Rule, Product Rule, and Quotient Rule. This solution provides a step-by-step approach, ideal for students in Grades 9-11 who are learning about logarithms and algebra.

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