6.4 Graphs of Logarithmic Functions
2 min read•june 24, 2024
Logarithmic functions are the inverse of exponential functions, with unique properties that set them apart. They have a of all positive real numbers and a range spanning all real numbers, making them versatile in various applications.
Graphing logarithmic functions involves understanding their key features, such as vertical asymptotes and x-intercepts. Transformations like shifts, stretches, and reflections allow us to manipulate these graphs, providing insights into their behavior and relationships.
Graphs of Logarithmic Functions
Domain and range of logarithms
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- of a logarithmic function includes all positive real numbers
- Argument of the (expression inside the ) must be greater than 0
- Logarithms undefined for non-positive numbers (0 and negative numbers)
- Range of a logarithmic function encompasses all real numbers
- As approaches 0 from the right, approaches negative infinity
- As approaches positive infinity, approaches positive infinity
Graphing logarithmic transformations
- Parent logarithmic function , where is the and
- Most common bases are 10 (common log) and (natural log, )
- Logarithmic functions are the inverse of exponential functions
- If , then
- This results in a of the over the line
- Transformations of logarithmic functions follow similar rules to other functions
- : shifts the graph up by units
- Horizontal shift: shifts the graph right by units
- Vertical stretch/:
- Stretches graph vertically by a factor of if
- Compresses graph vertically by a factor of if
- : reflects the graph over the -axis
Key features of logarithmic graphs
- Logarithmic functions have a at
- Graph approaches the but never touches or crosses it
- -intercept of a logarithmic function occurs when
- For the parent function , the -intercept is at
- Transformations can shift the -intercept
- Logarithmic functions do not have a -intercept, as is undefined
- The -axis serves as the vertical for the parent function
- of logarithmic functions:
- As , approaches negative infinity
- As , approaches positive infinity
Key Terms to Review (36)
Antilogarithm: The antilogarithm is the inverse operation of the logarithm. It is the process of finding the original number or value when given its logarithm. The antilogarithm is used to undo the effects of a logarithmic transformation and retrieve the original quantity or value.
Asymptote: An asymptote is a line or curve that a graph approaches but never touches. It represents the limit of a function's behavior as the input variable approaches a particular value. Asymptotes are an important concept in various mathematical topics, including rational expressions, functions, rational functions, exponential functions, logarithmic functions, and exponential and logarithmic models.
Base: The base is a fundamental component in various mathematical concepts, serving as a reference point or starting value. It is a crucial element in understanding exponents, exponential functions, logarithmic functions, and geometric sequences, among other topics.
Change of Base Formula: The change of base formula is a mathematical expression that allows for the conversion of logarithms from one base to another. This formula is particularly important in the context of logarithmic functions, their graphs, and the properties and equations involving logarithms and exponentials.
Change-of-base formula: The change-of-base formula is used to rewrite logarithms in terms of logs of another base, allowing for easier computation. It is commonly written as $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ where $b$ and $c$ are positive real numbers and $c \neq 1$.
Common logarithm: A common logarithm is a logarithm with base 10, often written as $\log_{10}(x)$ or simply $\log(x)$. It is commonly used in scientific calculations and when dealing with exponential growth or decay.
Common Logarithm: The common logarithm, also known as the base-10 logarithm, is a logarithmic function that expresses the power to which a base of 10 must be raised to obtain a given number. It is a fundamental concept in mathematics, with applications in various fields, including college algebra.
Compression: Compression refers to a transformation that reduces the distance between points in a graph. It often results in the graph appearing 'squeezed' either horizontally or vertically.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
End Behavior: The end behavior of a function refers to how the function behaves as the input variable approaches positive or negative infinity. It describes the limiting values or patterns that the function exhibits as it extends towards the far left and right sides of its graph.
Exponential function: An exponential function is a mathematical expression in the form $f(x) = a \cdot b^x$, where $a$ is a constant, $b$ is the base greater than 0 and not equal to 1, and $x$ is the exponent. These functions model growth or decay processes.
Exponential Function: An exponential function is a mathematical function in which the independent variable appears as an exponent. These functions exhibit a characteristic curve that grows or decays at a rate proportional to the current value, leading to rapid changes in output as the input increases.
Exponential growth: Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. This results in the function increasing rapidly over time.
Exponential Growth: Exponential growth is a pattern of change where a quantity increases at a rate proportional to its current value. This means the quantity grows by a consistent percentage over equal intervals of time, leading to rapid, accelerating growth. Exponential growth is a fundamental concept in mathematics and has applications across various fields, including biology, economics, and technology.
Horizontal reflection: A horizontal reflection is a transformation that flips a function's graph over the y-axis. It changes the sign of the x-coordinates of all points on the graph.
Horizontal stretch: A horizontal stretch is a transformation that scales a function's graph horizontally by multiplying the input values by a constant factor. If $0 < k < 1$, the graph stretches away from the y-axis.
Horizontal Stretch: A horizontal stretch is a transformation of a function that alters the x-coordinate of the graph, causing it to appear wider or narrower. This transformation affects the domain of the function, rather than the range.
Inverse function: An inverse function reverses the operation of a given function. If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Inverse Function: An inverse function is a function that reverses the operation of another function. It undoes the original function, mapping the output back to the original input. Inverse functions are crucial in understanding the relationships between different mathematical concepts, such as domain and range, composition of functions, transformations, and exponential and logarithmic functions.
Inverse Relationship: An inverse relationship is a relationship between two variables where an increase in one variable corresponds to a decrease in the other variable, and vice versa. This type of relationship is characterized by a negative correlation, where the variables move in opposite directions.
Ln: The natural logarithm, denoted as ln, is a logarithmic function that describes the power to which a base of e (approximately 2.718) must be raised to get a certain value. It is a fundamental mathematical concept that is closely related to exponential functions and is essential in understanding logarithmic functions, their graphs, and their properties.
Log: A logarithm is the exponent to which a base must be raised to get a certain number. It is a mathematical function that describes the power to which a fixed number, called the base, must be raised to produce a given value.
Logarithm: A logarithm is a mathematical function that describes the power to which a base number must be raised to get a certain value. It represents the exponent to which a base number must be raised to produce a given number. Logarithms are closely related to exponential functions and are essential in understanding topics such as logarithmic functions, graphs of logarithmic functions, exponential and logarithmic equations, and geometric sequences.
Logarithmic Properties: Logarithmic properties are the mathematical rules that describe the behavior and relationships between logarithmic functions. These properties provide a framework for understanding and working with logarithms, which are essential in the study of exponential and logarithmic functions.
Natural logarithm: The natural logarithm is the logarithm to the base $e$, where $e$ is an irrational and transcendental number approximately equal to 2.71828. It is commonly denoted as $\ln(x)$.
Natural Logarithm: The natural logarithm, denoted as $\ln(x)$, is a logarithmic function that represents the power to which the base $e$ must be raised to get the value $x$. The natural logarithm is a fundamental concept that underpins various topics in college algebra, including logarithmic functions, their graphs, properties, and applications in solving exponential and logarithmic equations, as well as modeling real-world phenomena.
Reflection: Reflection is a transformation that flips a graph over a specified axis, creating a mirror image. In algebra, this often involves reflecting exponential and logarithmic functions over the x-axis or y-axis.
Reflection: Reflection is a transformation of a function that creates a mirror image of the original function across a specified axis. This concept is fundamental in understanding the behavior and properties of various mathematical functions.
Vertical asymptote: A vertical asymptote is a line $x = a$ where a rational function $f(x)$ approaches positive or negative infinity as $x$ approaches $a$. It represents values that $x$ cannot take, causing the function to become unbounded.
Vertical Asymptote: A vertical asymptote is a vertical line that a graph of a function approaches but never touches. It represents the vertical limit of the function's behavior, indicating where the function's value becomes arbitrarily large or small.
Vertical shift: A vertical shift is a transformation that moves a graph up or down in the coordinate plane by adding or subtracting a constant to the function's output. It does not affect the shape of the graph, only its position.
Vertical Shift: Vertical shift refers to the movement of a graph or function up or down the y-axis, without affecting the shape or orientation of the graph. This transformation changes the y-intercept of the function, but leaves the x-intercepts and the overall shape unchanged.
X-intercept: The x-intercept is the point where a graph crosses the x-axis, where the y-coordinate is zero. It represents the solution(s) to an equation when $y = 0$.
X-Intercept: The x-intercept of a graph is the point where the graph of a function or equation intersects the x-axis, indicating the value of x when the function's output or the equation's value is zero. The x-intercept is a crucial concept in understanding the behavior and properties of various mathematical functions and equations.
Y-axis: The y-axis is the vertical axis in a rectangular coordinate system, which represents the dependent variable and is typically used to plot the values or outcomes of a function. It is perpendicular to the x-axis and provides a visual reference for the range of values a function can take on.