Logarithmic functions are the inverse of exponential functions, with unique properties and growth patterns. They're essential in modeling various real-world phenomena, from sound intensity to earthquake magnitude, making them crucial in many fields.
Graphing logarithmic functions involves understanding their domain, range, and key features like vertical asymptotes and intercepts. Transformations can shift, stretch, or reflect these graphs, allowing for a wide range of applications in problem-solving and data analysis.
Graphing Logarithmic Functions
Graphing logarithmic functions
General form of a logarithmic function y=logb(x)
b represents the base of the logarithm must be positive and not equal to 1 (b>0,b=1)
Common bases include 10 (common logarithm) and e (natural logarithm, approximately 2.718)
Graph of a logarithmic function is the of the corresponding exponential function across the line y=x
Logarithmic functions undergo transformations following the same rules as other functions
y=logb(x)+k moves the graph up by k units for k>0 and down by ∣k∣ units for k<0
y=logb(x−h) moves the graph right by h units for h>0 and left by ∣h∣ units for h<0
/ y=a⋅logb(x) stretches the graph vertically by a factor of ∣a∣ for ∣a∣>1 and compresses it for 0<∣a∣<1
Negative a values also reflect the graph across the x-axis
Horizontal stretch/compression y=logb(c⋅x) compresses the graph horizontally by a factor of ∣c∣ for ∣c∣>1 and stretches it for 0<∣c∣<1
Negative c values also reflect the graph across the y-axis
Domain and range of logarithms
Domain of a logarithmic function y=logb(x) includes all positive real numbers (0,∞)
Argument of the logarithm (value inside parentheses) must be greater than 0
Range of a logarithmic function encompasses all real numbers (−∞,∞)
As x approaches 0 from the right, y approaches negative infinity
As x approaches positive infinity, y approaches positive infinity
Key features of logarithmic graphs
Logarithmic functions have a vertical asymptote at x=0
Domain of the function excludes 0, and as x approaches 0 from the right, y approaches negative infinity
x-intercept of a logarithmic function y=logb(x) is always (1,0)
logb(1)=0 for any base b
Logarithmic function y=logb(x) does not have a y-intercept
Domain of the function excludes 0, so no value of x makes y=0
Logarithmic functions are always increasing for base b>1 and always decreasing for 0<b<1
Corresponding exponential function is always increasing for bases greater than 1 and decreasing for bases between 0 and 1
Growth patterns and transformations
Logarithmic functions exhibit logarithmic growth, which is slower than exponential growth
Exponential growth is characterized by rapid increase, while logarithmic growth slows down as x increases
Transformations of logarithmic functions include vertical and horizontal shifts, stretches, compressions, and reflections
The change of base formula allows conversion between logarithms of different bases: loga(x)=logb(a)logb(x)
Key Terms to Review (6)
Compression: Compression in algebra and trigonometry refers to the transformation that scales a graph towards an axis, making it narrower. This is achieved by multiplying the function by a constant factor.
Horizontal shift: A horizontal shift is a transformation that moves a graph left or right along the x-axis without changing its shape. It is represented by modifying the function as $f(x) \rightarrow f(x - h)$, where $h$ is the number of units shifted.
Parent function: A parent function is the simplest form of a family of functions that preserves the shape and orientation of the graph. Examples include $y = e^x$ for exponential functions and $y = \log_b(x)$ for logarithmic functions.
Reflection: Reflection is a transformation that flips a graph over a specific line, such as the x-axis or the y-axis, creating a mirror image. In algebra and trigonometry, it alters the sign of either the x-coordinates or y-coordinates of points on a graph.
Vertical shift: A vertical shift is a transformation that moves a graph up or down by adding or subtracting a constant to the function's output. This does not change the shape of the graph, only its position along the y-axis.
Vertical stretch: A vertical stretch is a transformation that scales a function's graph vertically by a factor greater than 1, making it appear taller. It multiplies the output value (y-coordinate) of each point by a given factor.