Power rule for logarithms

Definition

The power rule for logarithms states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number. Mathematically, $\log_b(a^c) = c \cdot \log_b(a)$ where $b$ is the base.

5 Must Know Facts For Your Next Test

1. The power rule can simplify complex logarithmic expressions.
2. It is applicable to any logarithmic base, not just base 10 or natural logs.
3. The rule helps in solving logarithmic equations involving exponents.
4. Combining it with other log rules (product and quotient) can further simplify expressions.
5. Knowing this rule is essential for calculus topics like differentiation and integration involving logs.

Review Questions

• What does the power rule for logarithms state?
• Simplify $\log_2(8^3)$ using the power rule.
• How would you transform $\log_3(x^5)$ using this property?

Related terms

Product Rule for Logarithms: States that the log of a product is equal to the sum of the logs: $\log_b(xy) = \log_b(x) + \log_b(y)$.

Quotient Rule for Logarithms: States that the log of a quotient is equal to the difference of the logs: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$.

Change of Base Formula: Allows you to change the base of a logarithm: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ where c is a new base.