Exponential and Logarithmic Functions

Graphs and Transformations of Exponential Functions

Exponential functions have the general form , where is a vertical stretch/compression, is the base, and is a vertical shift. Understanding their graphs and transformations is essential for analyzing their behavior.

• Y-intercept: The point where the graph crosses the y-axis (set ).

• Domain: The set of all possible -values (usually for basic exponentials).

• Range: The set of all possible -values (often or shifted by ).

• Horizontal Asymptote: The value approaches as or (typically ).

• Transformations: Shifts, reflections, and stretches/compressions from the parent function .

Example: For :

• Shift right 2 units, down 2 units.

• Domain:

• Range:

• Horizontal asymptote:

Exponential Growth and Decay Applications

Exponential functions model real-world phenomena such as population growth, radioactive decay, and compound interest. The general form is or .

• Growth: or

• Decay: or

• Applications: Savings accounts, atmospheric pressure, radioactive isotopes, population models.

Example: If , then , (growth rate).

Logarithmic Functions: Properties and Graphs

Logarithmic functions are the inverses of exponential functions. The general form is , where is the base.

• Domain: (for )

• Range:

• Vertical Asymptote: (or shifted by in )

• Transformations: Shifts, reflections, and stretches/compressions from the parent function

Example: For :

• Shift right 5 units, down 1 unit.

• Domain:

• Range:

• Vertical asymptote:

Evaluating and Manipulating Logarithms

Logarithms can be evaluated, expanded, or condensed using their properties:

• Product Rule:

• Quotient Rule:

• Power Rule:

• Change of Base Formula:

Example: because .

Expanding and Condensing Logarithmic Expressions

Logarithmic expressions can be rewritten using the properties above to either expand (write as a sum/difference of logs) or condense (combine into a single log).

• Expand:

• Condense:

Rewriting Equations: Exponential and Logarithmic Forms

Equations can be rewritten between exponential and logarithmic forms:

• Exponential to Logarithmic:

• Logarithmic to Exponential:

Example:

Table: Properties of Exponential and Logarithmic Functions

Additional info:

• Some problems involve sketching graphs and identifying transformations, which is a key skill for understanding function behavior.

• Applications include both continuous and discrete exponential models, as well as logarithmic equations for solving for time or rate in growth/decay problems.