A **logarithm** is the inverse operation of exponentiation, meaning it helps us find the exponent (or power) to which a base must be raised to obtain a given number.

### Key Points:
1. **Definition**: The logarithm of a number \( x \) with respect to a base \( b \) is written as \( \log_b(x) \). It means "what power must \( b \) be raised to, in order to get \( x \)?"
   \[
   \log_b(x) = y \quad \text{if and only if} \quad b^y = x
   \]
   For example, \( \log_2(8) = 3 \) because \( 2^3 = 8 \).

2. **Common Bases**:
   - **Base 10 (common logarithm)**: \( \log_{10}(x) \), usually written as \( \log(x) \) without the base.
     Example: \( \log(1000) = 3 \) because \( 10^3 = 1000 \).
   - **Base \( e \) (natural logarithm)**: \( \log_e(x) \), also written as \( \ln(x) \). The constant \( e \approx 2.718 \) is an irrational number.
     Example: \( \ln(e^2) = 2 \).

3. **Properties of Logarithms**:
   - **Product Rule**: \( \log_b(xy) = \log_b(x) + \log_b(y) \)
   - **Quotient Rule**: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
   - **Power Rule**: \( \log_b(x^n) = n \cdot \log_b(x) \)
   - **Change of Base Formula**: To convert between different bases:
     \[
     \log_b(x) = \frac{\log_c(x)}{\log_c(b)}
     \]
     where \( c \) can be any base (often base 10 or \( e \)).

4. **Inverse Relationship with Exponentials**: Since logarithms are the inverse of exponentiation, if \( b^y = x \), then \( \log_b(x) = y \).

### Example:
Let’s solve \( \log_3(81) \).
This asks: "To what power must 3 be raised to get 81?" Since \( 3^4 = 81 \), we find \( \log_3(81) = 4 \).

Logarithms are widely used in sciences, engineering, and finance, especially for handling very large or small numbers (like in pH levels or compound interest).

Would you like further details or have any questions?

### Follow-up Questions:
1. How do logarithms apply in real-world problems like interest rates or pH?
2. What is the difference between a natural logarithm and a common logarithm?
3. How does the change of base formula work with non-integer bases?
4. Can you explain logarithmic scales (e.g., Richter scale for earthquakes)?
5. How do logarithms simplify multiplication and division of large numbers?

**Tip**: Memorizing the logarithmic properties can help simplify complex expressions during calculations!

Learn the basics of logarithms, including their relationship with exponentiation, key properties, and example problems. This article covers logarithmic formulas, theorems, and common bases, suitable for students in Grades 9-12.

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