Logarithmic Functions

A logarithmic function is the inverse of an exponential function. The logarithm of a number is the power to which a given base must be raised to obtain that number. In other words, if y = log_b(x), then b^y = x. Here, 'b' is the base, 'x' is the argument, and 'y' is the logarithm.

Properties of Logarithmic Functions

1. Domain: The domain of a logarithmic function is the set of all positive real numbers.
2. Range: The range of a logarithmic function is the set of all real numbers.
3. Vertical Asymptote: The graph of a logarithmic function has a vertical asymptote at the equation x = 0.
4. Horizontal Asymptote: The graph of a logarithmic function does not have a horizontal asymptote.
5. One-to-One: Logarithmic functions are one-to-one functions, meaning each x-value corresponds to exactly one y-value.

Common Logarithmic Function

The common logarithmic function uses base 10. The equation for the common logarithmic function is y = log₁₀(x), often written as y = log(x).

Natural Logarithmic Function

The natural logarithmic function uses base e, where e is a constant approximately equal to 2.718. The equation for the natural logarithmic function is y = log_e(x), often written as y = ln(x).

Graphing Logarithmic Functions

To graph a logarithmic function, plot points and connect them smoothly. Remember the properties of logarithmic functions and asymptotes when graphing.

Study Guide

• Understand the concept of logarithms and their relationship with exponential functions.
• Learn the properties of logarithmic functions, including domain, range, and asymptotes.
• Practice solving logarithmic equations and inequalities.
• Master the graphs of common and natural logarithmic functions.
• Explore applications of logarithmic functions in real-world problems such as population growth and radioactive decay.

Remember to practice solving problems and graphing logarithmic functions to strengthen your understanding of the topic.