Logarithmic Equations

This page focuses on logarithmic equations, which are equations where the unknown variable appears as the argument of a logarithm. The page begins by introducing the elementary logarithmic equation:

Definition: An elementary logarithmic equation has the form log_a x = b, with the solution x = a^b, where x > 0.

The page then outlines three general types of logarithmic equations and their solving methods:

1. log_a f(x) = k
2. f(x) = log_a g(x)
3. a_1 log_a x + a_2 log_a x + a_3 = 0

Example: To solve log_2 x = 3, we get x = 2^3 = 8.

The solving process often involves substitution and solving resulting quadratic equations. The page emphasizes the importance of checking solutions against the domain restrictions of logarithms.

Highlight: Always remember the condition x > 0 when solving logarithmic equations, as logarithms are only defined for positive arguments.

Exponential and Logarithmic Inequalities

This final page covers exponential and logarithmic inequalities, which are crucial in solving many real-world problems involving growth and decay.

For exponential inequalities:

• a^(f(x)) > k becomes f(x) > log_a k for a > 1
• a^(f(x)) < k becomes f(x) < log_a k for 0 < a < 1

Example: Solve 2^x > 8. This becomes x > log_2 8 = 3.

For logarithmic inequalities:

• log_a f(x) > k becomes f(x) > a^k for a > 1
• log_a f(x) < k becomes 0 < f(x) < a^k for a > 1

Highlight: Always consider the domain restrictions when solving logarithmic inequalities: f(x) > 0.

The page also includes compound inequalities and systems of inequalities involving exponential and logarithmic functions.

Vocabulary: A compound inequality involves two or more inequalities combined with "and" or "or" statements.

This comprehensive overview of exponential and logarithmic inequalities completes the guide, providing students with the tools to tackle a wide range of problems in this important area of mathematics.