Intro to Logarithm Properties - Khan Academy

Overview

Learn about the properties of logarithms and how to use them to simplify and rewrite logarithmic expressions. Key properties include the product rule, quotient rule, and power rule.

Key Properties of Logarithms

1. Product Rule

• Formula: ( \log_b(MN) = \log_b(M) + \log_b(N) )
• Description: The logarithm of a product is the sum of the logarithms of its factors.
• Example:
  • If ( M = 4 ), ( N = 8 ), ( b = 2 ):
    • ( \log_2(48) = \log_2(4) + \log_2(8) )
    • Evaluated: ( 5 = 5 )
• Applications: Used to expand and condense logarithmic expressions.

2. Quotient Rule

• Formula: ( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) )
• Description: The logarithm of a quotient is the difference of the logarithms of the dividend and divisor.
• Example:
  • If ( M = 81 ), ( N = 3 ), ( b = 3 ):
    • ( \log_3\left(\frac{81}{3}\right) = \log_3(81) - \log_3(3) )
    • Evaluated: ( 3 = 3 )
• Applications: Used to expand and condense logarithmic expressions.

3. Power Rule

• Formula: ( \log_b(M^p) = p \cdot \log_b(M) )
• Description: The logarithm of a power is the exponent times the logarithm of the base of the power.
• Example:
  • If ( M = 4 ), ( p = 2 ), ( b = 4 ):
    • ( \log_4(4^2) = 2 \cdot \log_4(4) )
    • Evaluated: ( 2 = 2 )
• Applications: Used to expand and condense logarithmic expressions.

Important Notes

• All logarithms must have the same base when using these rules.
• The argument of a logarithm must be positive, and the base must be positive and not equal to 1.

Practice Problems

• Expand ( \log_2(3a) )
• Condense ( \log_5(2y) + \log_5(8) )

Challenge Problems

• Apply multiple properties in different scenarios to solve complex logarithmic equations.

Common Questions

• Why use logarithms in graphs?: Logarithms can simplify the relationships in exponential functions, making graphs easier to interpret.
• Purpose of using logarithms: Logarithms help solve equations involving exponential functions and are used in various scientific fields to deal with large numbers and multiplicative relationships.

Learn more by exploring further resources on Khan Academy's Algebra 2 course on properties of logarithms.