Logarithm Power Rule
The Power Rule of Logarithms states that when you have an exponent inside the logarithm, you can move that exponent in front as a multiplier. In mathematical terms, for any positive number \( a \), any real number \( c \), and a logarithm with base \( b \) (where \( b > 0 \) and \( b \neq 1 \)):
\[ \log_b(a^c) = c \cdot \log_b(a) \]Why Does This Work?
This rule comes from the definition of logarithms and the properties of exponents. Here’s a quick reasoning:
- Start with the definition: If \( \log_b(a) = y \), then by definition, \( b^y = a \).
- Apply the exponent \( c \) to both sides: \[ a^c = (b^y)^c = b^{yc} \]
- Take the logarithm base \( b \) of both sides: \[ \log_b(a^c) = \log_b(b^{yc}) \]
- Simplify using the inverse property of logarithms: \[ \log_b(b^{yc}) = yc = c \cdot \log_b(a) \]
Example
Suppose you want to simplify \( \log_{10}(1000^2) \):
- Recognize that \( 1000 = 10^3 \), so \( \log_{10}(1000) = 3 \).
- Apply the power rule: \[ \log_{10}(1000^2) = 2 \cdot \log_{10}(1000) = 2 \cdot 3 = 6 \]
Thus, \( \log_{10}(1000^2) = 6 \).
When to Use It
- Simplifying expressions: It helps in breaking down complex logarithmic expressions.
- Solving equations: It can turn multiplicative relationships into additive ones, making equations easier to solve.
- Calculus applications: It’s used frequently in differentiation and integration involving logarithmic functions.
The power rule is a fundamental tool in algebra and calculus, and mastering it can simplify many problems involving logarithms!
Latex
\log_b(a^c) = c \cdot \log_b(a)