Page 89 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

A. EXPONENTIAL AND LOGARITHMIC FUNCTIONS
OVERALL EXPECTATIONS
By the end of this course, students will:
1. demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;
2. identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically;
3. solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.
SPECIFIC EXPECTATIONS
1. Evaluating Logarithmic Expressions
By the end of this course, students will:
1.1 recognize the logarithm of a number to a given base as the exponent to which the base must be raised to get the number, recognize the operation of finding the logarithm to be the inverse operation (i.e., the undoing or reversing) of exponentiation, and evaluate simple logarithmic expressions
Sample problem: Why is it not possible to determine log10(– 3) or log20? Explain your reasoning.
1.2 determine, with technology, the approximate logarithm of a number to any base, including base 10 (e.g., by reasoning that log329 is between 3 and 4 and using systematic trial to determine that log329 is approximately 3.07)
1.3 make connections between related logarithmic and exponential equations (e.g., log5 125 = 3 can also be expressed as 53 = 125), and solve simple exponential equations by rewriting them in logarithmic form (e.g., solving 3x = 10 by rewriting the equation as log3 10 = x)
1.4 make connections between the laws of expo- nents and the laws of logarithms [e.g., use the statement 10a + b = 10a10b to deduce that log10 x + log10 y = log10 (xy)], verify the laws of logarithms with or without technology (e.g., use patterning to verify the quotient law for
logarithms by evaluating expressions such as log10 1000 – log10 100 and then rewriting the answer as a logarithmic term to the same base), and use the laws of logarithms to simplify and evaluate numerical expressions
By the end of this course, students will:
2.1 determine, through investigation with tech- nology (e.g., graphing calculator, spreadsheet) and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, increasing/decreasing behaviour) of the graphs of logarithmic func- tions of the form f(x) = logb x, and make con- nections between the algebraic and graphical representations of these logarithmic functions
Sample problem: Compare the key features of the graphs of f(x) = log2 x, g(x) = log4 x, and h(x) = log8 x using graphing technology.
2.2 recognize the relationship between an expo- nential function and the corresponding loga- rithmic function to be that of a function and its inverse, deduce that the graph of a loga- rithmic function is the reflection of the graph of the corresponding exponential function in the line y = x, and verify the deduction using technology
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
2. Connecting Graphs and Equations of Logarithmic Functions
87
Advanced Functions
MHF4U