Graphs of Logarithmic Functions
Investigate the graphical representation of logarithmic functions, y = log_a(x), identifying key features such as the vertical asymptote, domain, range, and intercepts.
TL;DR:Unlock the visual side of logarithms by exploring their graphs. This topic builds a crucial bridge between the algebraic rules of logs and their behaviour as functions.
About This Topic
This topic delves into the graphical representation of logarithmic functions, a core component of the Year 11 Mathematical Methods and equivalent courses within the Australian Curriculum. It builds directly upon students' prior understanding of index laws and exponential functions, formalising the inverse relationship between y = a^x and y = log_a(x). The focus is on developing a deep conceptual understanding of the graph's key features: the vertical asymptote, the domain restriction to positive real numbers, the range of all real numbers, and the x-intercept.
By exploring these graphs, students gain a visual intuition for how logarithms behave, particularly how the base 'a' influences the rate of growth. This is crucial for later topics involving calculus with logarithmic functions and for solving logarithmic equations and inequalities. Mastery of these graphical concepts provides a solid foundation for applications in various scientific fields, such as chemistry (pH scale) and seismology (Richter scale), which are often explored in senior science subjects.
Key Questions
- Analyse the relationship between the graph of y = a^x and y = log_a(x) with respect to the line y = x.
- Explain why the graph of a logarithmic function has a vertical asymptote but no horizontal asymptote.
- Compare the graphs of y = log_2(x) and y = log_10(x), noting similarities and differences in their shape and rate of increase.
Learning Objectives
- Identify the key features of y = log_a(x), including the vertical asymptote, domain, range, and x-intercept.
- Sketch graphs of logarithmic functions for different bases, including transformations.
- Explain the graphical relationship between exponential and logarithmic functions as inverses.
- Analyse the effect of the base on the shape and rate of increase of a logarithmic graph.
- Compare and contrast the graphs of different logarithmic functions on the same axes.
Key Vocabulary
| Logarithm | The exponent or power to which a fixed number, the base, must be raised to produce a given number. |
| Vertical Asymptote | A vertical line that the graph of a function approaches but never touches or crosses. |
| Domain | The complete set of possible input values (x-values) for which the function is defined. For y = log_a(x), the domain is (0, ∞). |
| Range | The complete set of possible output values (y-values) of a function. For y = log_a(x), the range is all real numbers. |
| Inverse Function | A function that reverses the action of another. Graphically, its graph is a reflection of the original function in the line y = x. |
Watch Out for These Misconceptions
Common MisconceptionThe graph of a log function can have a y-intercept.
What to Teach Instead
The basic function y = log_a(x) never touches or crosses the y-axis, as its domain is x > 0. The y-axis (x=0) is a vertical asymptote. Only a horizontally translated function, like y = log_a(x+c) where c > 0, can have a y-intercept.
Common MisconceptionYou can't have a negative answer for a logarithm.
What to Teach Instead
The output of a logarithm (the y-value) can certainly be negative. This occurs when the input value (x) is between 0 and 1. The restriction is on the input: you cannot take the logarithm of a negative number or zero.
Common MisconceptionThe base of the logarithm doesn't really change the graph's shape.
What to Teach Instead
While all graphs of y = log_a(x) for a > 1 are similar, the base 'a' significantly impacts the graph's steepness. A smaller base results in a steeper graph that increases more rapidly for x > 1.
Active Learning Ideas
See all activities→Gallery Walk
Inverse Function Mirror
Students first plot a familiar exponential function, like y = 2^x. They then swap the x and y coordinates for each point and plot the new points, discovering the shape of y = log_2(x) and its reflection across the line y = x.
Gallery Walk
Asymptote Investigator
Using a graphing calculator or online tool, students investigate the value of y = log_10(x) as x gets closer and closer to zero (e.g., x=0.1, 0.01, 0.001). They record their findings to conclude why the y-axis is a vertical asymptote.
Gallery Walk
The Base Race
In small groups, students graph y = log_2(x), y = log_e(x), and y = log_10(x) on the same set of axes. They then compare the steepness of the graphs and discuss which function increases the fastest and why.
Real-World Connections
- Measuring earthquake magnitude on the Richter scale, where each whole number increase represents a tenfold increase in measured amplitude.
- Quantifying the acidity of a solution using the pH scale, which is based on the negative logarithm of hydrogen ion concentration.
- Calculating the loudness of sound in decibels (dB), a logarithmic scale that compares sound pressure to a reference level.
- Modelling human perception, as our senses of sight and hearing perceive brightness and loudness logarithmically.
- Used in finance to determine the time required for an investment to grow to a certain value with compound interest.
Assessment Ideas
An 'exit ticket' task where students are given a logarithmic function and must sketch its graph, labelling the x-intercept and the equation of the vertical asymptote.
A test question that requires students to graph a transformed logarithmic function, such as y = 2log_10(x - 1), and state its domain and range.
A 'pair and compare' activity where students each graph a different function (e.g., y=log_2(x) and y=log_4(x)) and then explain the similarities and differences to their partner.
Frequently Asked Questions
Why is the domain of y = log_a(x) only positive numbers?
What is the difference between log(x) and ln(x) on my calculator?
How are logarithmic graphs and exponential graphs related?
Planning templates for Mathematics
5E Model
The 5E Model structures lessons through five phases (Engage, Explore, Explain, Elaborate, and Evaluate), guiding students from curiosity to deep understanding through inquiry-based learning.
Unit PlannerMath Unit
Plan a multi-week math unit with conceptual coherence: from building number sense and procedural fluency to applying skills in context and developing mathematical reasoning across a connected sequence of lessons.
RubricMath Rubric
Build a math rubric that assesses problem-solving, mathematical reasoning, and communication alongside procedural accuracy, giving students feedback on how they think, not just whether they got the right answer.
More in Exponential and Logarithmic Functions
Review of Index Laws and Rational Exponents
Revisit and consolidate your understanding of the index laws for integer and rational exponents, which form the foundation for exponential functions.
8 methodologies
The Exponential Function and its Graph
Explore the family of exponential functions, y = a^x, and understand how the base 'a' affects the shape and key features of their graphs, including asymptotes and intercepts.
8 methodologies
Logarithms as Inverses
Discover logarithms as the inverse operation of exponentiation. Understand the fundamental relationship that log_a(x) = y is equivalent to a^y = x.
8 methodologies
The Laws of Logarithms
Master the fundamental laws of logarithms, including the product, quotient, and power rules, to simplify and manipulate logarithmic expressions.
8 methodologies
Solving Exponential and Logarithmic Equations
Develop algebraic strategies to solve equations involving exponential or logarithmic terms by applying index laws, logarithm laws, and the inverse relationship between them.
8 methodologies
Applications of Exponential and Logarithmic Models
Apply your knowledge to model and solve real-world problems involving exponential growth and decay, such as compound interest, population dynamics, and radioactive decay.
8 methodologies