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:

1. Start with the definition: If ( \log_b(a) = y ), then by definition, ( b^y = a ).
2. Apply the exponent ( c ) to both sides: $$a^c = (b^y)^c = b^{yc}$$
3. Take the logarithm base ( b ) of both sides: $$\log_b(a^c) = \log_b(b^{yc})$$
4. 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) ):

1. Recognize that ( 1000 = 10^3 ), so ( \log_{10}(1000) = 3 ).
2. 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)