Page 90 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, University Preparation
3. Solving Exponential and Logarithmic Equations
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
Sample problem: Give examples to show that the inverse of a function is not necessarily a function. Use the key features of the graphs of logarithmic and exponential functions to give reasons why the inverse of an exponential function is a function.
2.3 determine, through investigation using technol- ogy, the roles of the parameters d and c in functions of the form y = log10(x – d) + c and the roles of the parameters a and k in func- tions of the form y = alog10(kx), and describe these roles in terms of transformations on the graph of f(x) = log10x (i.e., vertical and horizontal translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)
Sample problem: Investigate the graphs of f(x) = log10(x) + c, f(x) = log10(x – d),
f(x) = alog10x, and f(x) = log10(kx) for
various values of c, d, a, and k, using technol-
ogy, describe the effects of changing these parameters in terms of transformations, and make connections to the transformations of other functions such as polynomial functions, exponential functions, and trigonometric functions.
2.4 pose problems based on real-world applica- tions of exponential and logarithmic functions (e.g., exponential growth and decay, the Richter scale, the pH scale, the decibel scale), and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation
Sample problem: The pH or acidity of a solu- tion is given by the equation pH = – logC, where C is the concentration of [H+] ions in multiples of M = 1 mol/L. Use graphing software to graph this function. What is the change in pH if the solution is diluted from a concentration of 0.1M to a concentration of 0.01M? From 0.001M to 0.0001M? Describe the change in pH when the concentration of
any acidic solution is reduced to 1 of its 10
original concentration. Rearrange the given
equation to determine concentration as a function of pH.
By the end of this course, students will:
3.1 recognize equivalent algebraic expressions involving logarithms and exponents, and simplify expressions of these types
Sample problem: Sketch the graphs of f(x) = log10(100x) and g(x) = 2 + log10x, compare the graphs, and explain your findings algebraically.
3.2 solve exponential equations in one variable by determining a common base (e.g., solve 4x = 8x + 3 by expressing each side as a power of 2) and by using logarithms (e.g., solve
4x = 8x + 3 by taking the logarithm base 2
of both sides), recognizing that logarithms base 10 are commonly used (e.g., solving
3x = 7 by taking the logarithm base 10 of both sides)
Sample problem: Solve 300(1.05)n = 600 and 2x+2 –2x =12eitherbyfindingacommon base or by taking logarithms, and explain your choice of method in each case.
3.3 solve simple logarithmic equations in one variable algebraically [e.g., log3(5x + 6) = 2, log10(x + 1) = 1]
3.4 solve problems involving exponential and logarithmic equations algebraically, includ- ing problems arising from real-world applications
Sample problem: The pH or acidity of a solu- tion is given by the equation pH = – logC, where C is the concentration of [H+] ions in multiples of M = 1 mol/L. You are given a solution of hydrochloric acid with a pH of 1.7 and asked to increase the pH of the solution by 1.4. Determine how much you must dilute the solution. Does your answer differ if you start with a pH of 2.2?
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