A **logarithm** is a mathematical concept that helps you solve equations involving exponentiation. It answers the question: "To what exponent must a certain base be raised to produce a given number?"

### The General Form
For a given base \( b \), a logarithm can be written as:
\[
\log_b(a) = x
\]
This means that \( b^x = a \).

### Common Types of Logarithms
1. **Common Logarithm (Base 10):** \(\log_{10}(x) \) is often written simply as \( \log(x) \). It's widely used in scientific calculations.
2. **Natural Logarithm (Base \(e\)):** \( \log_e(x) \), written as \( \ln(x) \), where \( e \approx 2.71828 \). It's fundamental in calculus and natural growth processes.
3. **Binary Logarithm (Base 2):** \( \log_2(x) \), frequently used in computer science.

### Logarithm Properties
1. **Product Rule:** 
   \[
   \log_b(xy) = \log_b(x) + \log_b(y)
   \]
2. **Quotient Rule:** 
   \[
   \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)
   \]
3. **Power Rule:** 
   \[
   \log_b(x^c) = c \cdot \log_b(x)
   \]
4. **Change of Base Formula:** 
   \[
   \log_b(a) = \frac{\log_c(a)}{\log_c(b)}
   \]
   You can convert logarithms from one base to another using this formula.

Would you like more details or have any specific questions about logarithms?

### Here are 5 related questions to deepen your understanding:
1. What is the value of \( \log_{10}(1000) \)?
2. How does the natural logarithm differ from other logarithms?
3. Can you convert \( \log_2(8) \) to a logarithm with base 10?
4. What is the inverse operation of a logarithm?
5. How can logarithms be used to solve exponential equations?

**Tip:** Understanding the properties of logarithms is key to simplifying complex logarithmic expressions!

Explore the concept of logarithms and their key properties, including product, quotient, and power rules. This page provides a detailed explanation of logarithms with step-by-step solutions to common logarithmic problems, ideal for students in Grades 9-12.

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