Pre-Calculus 12 Pacing Guide - This pacing guide replaces the previous yearly plan. It has been updated to reflect removed outcomes and provide flexibility for responsive instruction.
Pre-calculus 12 Desmos Activity Collection - A collection of online student Desmos activities organized by unit.
EAL Support - Desmos offers a free suite of math software tools, including the Desmos Graphing Calculator and Scientific Calculator, as well as free digital classroom activities. Click on the globe in the tool bar to access the site in other languages.
RF08 - Although the term argument is not emphasized in Pre-Calculus 12 (McAskill et al. 2012), it is suggested that you use this term when discussing logarithms
RF09 - Students are not responsible for determining the equation of a logarithmic function, given the graph. However, given the graph of a logarithmic function, they should identify the function from a list of options.
RF07 Students will be expected to demonstrate an understanding of logarithms. [CN, ME, R]
RF07.01 Explain the relationship between logarithms and exponents.
RF07.02 Express a logarithmic expression as an exponential expression and vice versa.
RF07.03 Determine, without technology, the exact value of a logarithm, such as log₂ 8 and ln e.
RF07.04 Estimate the value of a logarithm, using benchmarks, and explain the reasoning (e.g., since log₂ 8 = 3 and log₂ 16 = 4, log₂ 9 is approximately equal to 3.1).
RF08 Students will be expected to demonstrate an understanding of the product, quotient, and power laws of logarithms. [C, CN, R, T]
RF08.01 Develop and generalize the laws for logarithms, using numeric examples and exponent laws.
RF08.02 Derive each law of logarithms.
RF08.03 Determine, using the laws of logarithms, an equivalent expression for a logarithmic expression.
RF08.04 Determine, with technology, the approximate value of a logarithmic expression, such as log₂ 9 and ln 10.
RF09 Students will be expected to graph and analyze exponential and logarithmic functions. [C, CN, T, V]
RF09.01 Sketch, with or without technology, a graph of an exponential function of the form y = aˣ , a > 0.
RF09.02 Identify the characteristics of the graph of an exponential function of the form y = aˣ , a > 0, including the domain, range, horizontal asymptote and intercepts, and explain the significance of the horizontal asymptote.
RF09.03 Sketch the graph of an exponential function by applying a set of transformations to the graph of y = aˣ , a > 0, and state the characteristics of the graph.
RF09.04 Sketch, with or without technology, the graph of a logarithmic function of the form y = log_b x, b > 1.
RF09.05 Identify the characteristics of the graph of a logarithmic function of the form y = log_b x, b > 1, including the domain, range, vertical asymptote and intercepts, and explain the significance of the vertical asymptote.
RF09.06 Sketch the graph of a logarithmic function by applying a set of transformations to the graph of y = log_b x, b > 1, and state the characteristics of the graph.
RF09.07 Demonstrate, graphically, that a logarithmic function and an exponential function with the same base are inverses of each other.
RF10 Students will be expected to solve problems that involve exponential and logarithmic equations. [C, CN, PS, R]
RF10.01 Determine the solution of an exponential equation in which the bases are powers of one another.
RF10.02 Determine the solution of an exponential equation in which the bases are not powers of one another, using a variety of strategies.
RF10.03 Determine the solution of a logarithmic equation, and verify the solution.
RF10.04 Explain why a value obtained in solving a logarithmic equation may be extraneous.
RF10.05 Solve a problem that involves exponential growth or decay.
RF10.06 Solve a problem that involves the application of exponential equations to loans, mortgages, and investments.
RF10.07 Solve a problem that involves logarithmic scales, such as the Richter scale and the pH scale.
RF10.08 Solve a problem by modelling a situation with an exponential or a logarithmic equation.
Additional Resources and Activities for RF07 and RF08 (understanding logarithms and logarithm laws):
Logarithms Speed Dating - *Updated June 2025* - Students take turns pairing up and quizzing and coaching each other through solving the problem on the card. When both students have solved both cards, the students trade cards and find a new partner (from Sara Carter's Blog M+A+T+H=Love).
Quick Thoughts on Logarithms - A number of activities regarding how to introduce the concept of logarithms in your class. From Sara VanDerWerf
Logarithms Tarsia puzzle and a larger sized Tarsia puzzle.- Two jigsaw puzzles to practice logarithms. and also talks about the Tarsia program used to create them. Created by Chris Hunter talks about the Tarsia program used to create these puzzles on his blog.
Logarithms Clothesline - Ask students to design logarithm questions given a set of answers. The next day, use these expressions to have students sort them and place them in the correct order on a hanging clothesline. Created by Caitlyn Gironda.
Properties of Logs Circuit - A self-checking sheet for students to practice using laws of logarithms to simplify or expand expressions. From Girl Math
Logarithm War - Students take turns mentally evaluating and comparing the values of log expressions using the card game War.
Exponents & Logarithms the MTBoS way - A number of activities to teach exponents and logarithms. From John Rowe.
Logarithms Open - Middle - *Added March 2025* (From: How I Teach Maths Blog by John Rowe). Create correct logarithmic expressions using the values 0 - 9 only once.
Note: Euler's number, e, and the natural logarithm, ln(x), do not show up in the textbook expect for the C3 Mini-Lab on page 382. They are mentioned in the curriculum guide with the following, "Students may have not been introduced to the number e, Euler’s number, prior to this unit. This constant can be used to describe naturally occurring constant growth rates. ( e is approximately 2.718281828 ) Natural logarithms have a base of e and can be written as y = log_e (x) or in its more common abbreviated form simply as y = ln(x) . Students should also be familiar with this convention. "
Calculating e by Hand - Pick a random number between 0 and 1. Do this again and add the second number to the first number. Keep doing this until the sum of the numbers exceeds 1. What's the expected value of the number of random numbers needed to accomplish this? On average, this will take e picks. You can turn this into an investigation by rolling dice in a variety of ways.
e (Euler's Number) video - Dr James Grime discusses "e" - the famed Euler's Number. From Numberphile
Additional Resources and Activities for RF09 (graph log and exponential functions):
Folding Paper More than Seven Times YouTube video - A popular myth exists that one can only fold a single piece of paper in half, seven, maybe eight times at most. High school student Britney Gallivan took on this challenge and is now has a Guinness World Record. Mythbusters also took on this myth. There are lots of classroom activities using this idea.
Matryoshka Dolls - What do you notice about these dolls? What do you wonder? After discussion, reveal tallest doll 21.25 inches, smallest .12 inches, 51 total dolls. What’s the decay rate? The image shows the largest set of matryoshka dolls in the world is a 51-piece set hand-painted by Youlia Bereznitskaia of Russia, completed in 2003. The tallest doll in the set measures 53.97 centimetres (21.25 in); the smallest, 0.31 centimetres (0.12 in). From Mike Larson
Exponentials Polygraph Desmos Activity - This Custom Polygraph is designed to spark vocabulary-rich conversations about exponentials, including how they differ from linear functions. Key vocabulary that may appear in student questions includes: increasing, decreasing, intercept, rate, asymptote, and curve.
Marbleslides: Exponentials Desmos Activity - In this delightful and challenging activity, students will transform exponential functions so that the marbles go through the stars. Students will test their ideas by launching the marbles, and have a chance to revise before trying the next challenge.
Additional Resources and Activities for RF10 (solve log and exponential equations):
Exponent Puzzles - A review of the relationship between exponents and logarithms. Includes a series of 3 sets of problems of increasing complexity for exponential equations. They are all set to print and put into dry erase sleeves for students to work with. From Amy Gruen.
Logarithms Square Puzzle - *Updated June 2025* - Students solve simple log equations, and align questions and answers to form a square (from Sara Carter's Blog M+A+T+H=Love)
Laws of Logarithms Add Em Up activity - A practice activity for laws of logarithms. Students work in groups to solve a set of problems. If they are not all correct, they have to problem solve to find out which answer(s) are wrong. More back ground on this type of self-checking practice activity can be found on Sara VanDerWerf's blog site. A similar activity that also includes exponential equations, Sum of 3, can be found at Girl Math.
Logarithmic Equations Secret Message activity - On this worksheet, students solve six logarithmic equations. The answers are used to create a keyword. This keyword can then be used to decrypt a secret message that was created with a Vigenère cipher.
Growing Blocks Desmos Activity - In this Desmos activity, students discover how to write exponential growth functions given a starting value other than 1.
Solving Exponential and Logarithmic Equations Drag and Drop - A google form worksheet for students to practice solving logarithmic and exponential equations. From Girl Math
Creating a Need for Logarithms with Zombies - Students are turning into Zombies at an exponential rate! How many hours until the entire student body is ALL Zombies? Solving this problem without logarithms will take some persistence and trial and error. Once students have a good idea of the solution, show them how to use logarithms to find the exact answer. From Julie Reulbach.
The Triangle of Power video - In math, exponents, logarithms, and roots all circle around the same idea, but the notation for each varies radically. Grant Sanderson describes the triangle of power as an alternate notation. Also a Which One Doesn't Belong based on the triangle of power from Luke Walsh. Created by 3Blue1Brown.
Plotting the Planets - In this resource, students use logarithms to explore real data, investigate relationships and produce modelling formulae.