Logarithmic Functions
Definition
A logarithm is the inverse function to exponentiation. It answers the question: to what exponent must the base be raised to obtain a given number?
For base \( b \) and number \( x \), the logarithm is defined as:
\[ \log_b(x) = y \quad \text{if and only if} \quad b^y = x \]
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: \[ \log_b(x^k) = k \cdot \log_b(x) \]
- Change of Base Formula: \[ \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \]
Common Logarithms
The most common logarithms are:
- Common Logarithm: \[ \log_{10}(x) \text{ or simply } \log(x) \]
- Natural Logarithm: \[ \log_e(x) \text{ or } \ln(x) \]
Examples
Solve the following logarithmic equations:
- Solve: \[ \log_2(8) = ? \] Since \( 2^3 = 8 \), we have: \[ \log_2(8) = 3 \]
- Solve: \[ \log_{10}(1000) = ? \] Since \( 10^3 = 1000 \), we have: \[ \log_{10}(1000) = 3 \]
- Simplify: \[ \log_b(b^5) = ? \] By the power rule: \[ \log_b(b^5) = 5 \]