Logarithms and Logarithmic Functions

This page provides a comprehensive overview of logarithms and logarithmic functions, covering their definitions, properties, and applications in mathematics.

Definition: A logarithm is the power to which a base number must be raised to yield a given number. It is written as log_a$x$ = y, where 'a' is the base, 'x' is the argument, and 'y' is the result.

The page begins by introducing the concept of logarithms, emphasizing the relationship between the base, argument, and result. It then focuses on common logarithms, which use base 10.

Highlight: Common logarithms, also known as base 10 logarithms, are denoted simply as 'log' instead of 'log_10'.

Examples of common logarithms are provided:

• log 10 = 1
• log 100 = 2

The logarithmic function is then introduced and defined.

Definition: A logarithmic function is a function of the form f$x$ = log_a$x$, defined for x > 0, where 'a' is the base of the logarithm.

A graph of the logarithmic function is included, illustrating its shape and key characteristics.

The page also covers important properties of logarithms:

1. Logarithm of a product: log_a$b * c$ = log_a$b$ + log_a$c$
2. Logarithm of a quotient: log_a(b / c) = log_a$b$ - log_a$c$
3. Logarithm of a power: log_a$b^r$ = r * log_a$b$
4. Change of base formula: log_a$b$ = log_c$b$ / log_c$a$

Example: log_a(1) = 0 for any base 'a', as a^0 = 1

These properties are fundamental for solving complex logarithmic equations and simplifying expressions involving logarithms.

The page concludes by emphasizing that for all logarithmic operations, the base 'a' must be positive and not equal to 1, and the argument 'b' must be strictly positive.

This comprehensive overview provides students with a solid foundation for understanding and working with logarithms and logarithmic functions, preparing them for more advanced mathematical concepts and applications.