Power rule for logarithms

Definition

The power rule for logarithms states that $\log_b(x^r) = r \cdot \log_b(x)$, where $b$ is the base of the logarithm. This property allows you to move the exponent in a logarithmic expression to the front as a multiplier.

5 Must Know Facts For Your Next Test

1. The power rule can simplify complex logarithmic expressions involving exponents.
2. It applies to any base $b$, not just common bases like 10 or $e$.
3. To use the power rule effectively, ensure that the argument of the logarithm is positive and real.
4. This rule is often used in conjunction with other logarithmic properties such as the product and quotient rules.
5. Understanding this rule is crucial for solving exponential equations by converting them into logarithmic form.

Review Questions

• How would you simplify $\log_2(8^3)$ using the power rule?
• Explain why $\log_b(x^r) = r \cdot \log_b(x)$ is valid for any positive real number $x$ and any real number $r$.
• What are some potential pitfalls when applying the power rule to negative or zero arguments?

Related terms

Logarithm: A mathematical function that determines how many times one number must be multiplied by itself to obtain another number, denoted as $\log_b(x)$.

Product Rule for Logarithms: $\log_b(MN) = \log_b(M) + \log_b(N)$, which states that the logarithm of a product is equal to the sum of the logarithms.

Quotient Rule for Logarithms: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$, indicating that the logarithm of a quotient is equal to the difference of the logarithms.