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!