Lecture on Logarithms

Key Definitions

• Base: The number being raised to an exponent.
• Exponent: The power to which the base is raised.

Important Properties and Rules

• Product Rule: ( \log_b (a \cdot b) = \log_b a + \log_b b )
• Quotient Rule: ( \log_b \left( \frac{a}{b} \right) = \log_b a - \log_b b )
• Power Rule: ( \log_b (a^n) = n \log_b a )

Key Points

• Logarithms with base 10 are often used in problems.
• Be familiar with changing the base and manipulating exponents.

Problem Solving Steps

Example 1

• Problem: Evaluate ( 25^x
• Write down: ( 25 = 5^2 )
• Expression becomes ( (5^2)^x
• Solve for x.

Example 2

• Problem: Solve the equation ( 3^x = 5^{x-2} )
• Steps:
  1. Take the logarithm of both sides: ( \log (3^x) = \log (5^{x-2}) )
  2. Apply Power Rule: ( x \log 3 = (x-2) \log 5 )
  3. Distribute: ( x \log 3 = x \log 5 - 2 \log 5 )
  4. Combine like terms to solve for x.

Example 3

• Problem: Simplify ( \frac{\log 25 + \log 3 - \log 3}{\log X} )
  1. Use log properties to simplify numerator.
  2. Compare and solve for X.

Example 4

• Problem: Solve for ( \frac{1}{x} - \frac{1}{y} ) when ( 2.3^x = 0.23^y ) with logarithms.
• Steps:
  1. Express both equations using logarithms.
  2. Divide the equations.
  3. Combine and solve for the variables.

Homework

• Solve given logarithmic problems and submit answers in the comment section.

Conclusion

• Techniques: Understand and apply logarithm properties for solving equations.
• Practice: Gain proficiency through problem-solving and practice.

We'll continue with more strategies and examples in the next session. Until then, keep practicing!