2.11 Logarithmic Functions
Introduction to Logarithmic Functions
Overview of topics: shifting, graphing, intervals of increase, domain, and range of logarithmic functions.
Exponential Functions
General Information
Exponential graphs can be represented by the equation y = B^x.
Behavior based on base B:
Increasing: When B > 1
Decreasing: When 0 < B < 1.
Key Features
X-Intercept: Exponential functions do not cross the x-axis (no x-intercept).
Y-Intercept: Always at (0, 1) regardless of B.
Asymptotes
Asymptote for exponential graphs is a horizontal line at y = 0.
Domain and Range
Domain of exponential functions: All real numbers (−∞, +∞).
Range: For increasing functions, [0, +∞); for decreasing functions, (0, +∞).
Logarithmic Functions
Key Features
Logarithms are the inverse of exponential functions. Their key characteristics are swapped:
X-Intercept: (1, 0)
Y-Intercept: No y-intercept because logarithmic functions never cross the y-axis.
Asymptotes
Asymptote for logarithmic functions is a vertical line at x = 0.
Domain and Range
Domain: For logarithmic functions, the domain is (0, +∞).
Range: All real numbers (−∞, +∞).
Characteristics of Logarithmic Graphs
Behavior
If B > 1, the logarithmic function is increasing.
If 0 < B < 1, the logarithmic function is decreasing.
The growth of logarithmic functions slows down infinitely.
Graphing Basics
Logarithmic graphs approach but never touch the vertical asymptote.
Points plotted will show (1, 0) on the log graph, mirroring characteristics of the exponential graph at (0, 1).
Graph Comparisons
Example of log base 2: As x increases, the output doubles in behavior and becomes less steep as x grows larger.
Comparison of log and exponential graphs shows their inverses:
For exponential: move right by 1 increases y rapidly.
For logarithmic: requires larger moves in x to achieve the same increase in y.
Shifting Logarithmic Functions
Horizontal and Vertical Shifts
Horizontal Shift: x → (x + 4) shifts left by 4; affecting domain.
Vertical Shift: Logarithmic function of log_2(x) + 4 shifts up by 4; does not affect domain.
Summary of Shifts
Shifting affects where the function starts but does not change the range of all real numbers.
Horizontal shifts impact domain; vertical shifts primarily influence the y-values only.
Dilation and Reflection in Logarithmic Functions
Vertical Dilation
Represented as a multiplier on the function outside; affects the steepness of the curve but not the asymptote.
Horizontal Dilation
Horizontal dilation represented as a multiplier inside; compresses or expands the graph horizontally, affecting growth rates.
Reflections
Vertical Reflection: Negative multiplier outside flips the graph upside down, not affecting the range.
Horizontal Reflection: Negative multiplier inside affects the function's domain by altering where it intersects or approaches the asymptote.
Examining Asymptotes, Domain, and Range
Sketching Functions
Understanding the asymptote assists in identifying the domain.
Typically, if identifying an asymptotic function, the domain ends at that point and begins at negative infinity or a specified number.
End Behavior
Writing limits as x approaches a particular number, observing whether it approaches negative or positive infinity.
Practical Application and Discussion
Importance of understanding calculator functionality when graphing logarithmic functions for observational accuracy.
Encouragement to visualize, sketch and engage with function transformations to solidify understanding.