Lesson Notes By Weeks and Term v5 - Grade 12
Exponential and logarithmic functions – Week 6 focus
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Exponential and logarithmic functions – Week 6 focus
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Subject: Mathematics
Class: Grade 12
Term: 1st Term
Week: 6
Theme: General lesson support
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Exponential and logarithmic functions are fundamental tools in mathematics with far-reaching applications in various fields, including finance, science, and engineering. Understanding these functions is crucial for modelling growth and decay phenomena, which are prevalent in South African contexts, such as population dynamics, compound interest on investments (like stokvels), and the depreciation of assets. This week's focus builds upon the foundational knowledge of exponents from previous grades, introducing logarithms as the inverse operation and exploring their properties and graphs.
2.1 Exponential Functions: An exponential function is defined as f(x) = a x , where a is a positive real number not equal to 1 (i.e., a > 0 and a ≠ 1). a is called the base, and x is the exponent.
Key Features: Domain: All real numbers (ℝ).
Range: y > 0 if a > 1 (increasing function); y > 0 if 0 x + q: Vertical translation by q units. y = a x+p : Horizontal translation by p units to the left. y = -a x : Reflection about the x-axis. y = a -x : Reflection about the y-axis.
Example 1: Sketch the graph of y = 2 x .
Create a table of values: | x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | | :---- | :----- | :----- | :----- | :-: | :-: | :-: | :-: | | y = 2 x | 1/8 = 0.125 | 1/4 = 0.25 | 1/2 = 0.5 | 1 | 2 | 4 | 8 | Plot the points and draw a smooth curve. The graph has a horizontal asymptote at y =
0. Example 2: Sketch the graph of y = (1/2) x .
Create a table of values: | x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | | :---- | :-- | :-- | :-- | :-: | :----- | :----- | :----- | | y = (1/2) x | 8 | 4 | 2 | 1 | 1/2 = 0.5 | 1/4 = 0.25 | 1/8 = 0.125 | Plot the points and draw a smooth curve. The graph has a horizontal asymptote at y = 0. 2.2 Logarithmic Functions: The logarithm is the inverse operation to exponentiation. If a y = x, then log a (x) = y, where a > 0, a ≠ 1, and x > 0. a is the base of the logarithm.
Key Features: Domain: x >
0. Range: All real numbers (ℝ).
Vertical Asymptote: The y-axis (x = 0). x-intercept: (1, 0).
Transformations: y = log a (x) + q: Vertical translation by q units. y = log a (x+p): Horizontal translation by p units to the left. y = -log a (x): Reflection about the x-axis. y = log a (-x): Reflection about the y-axis.
Example 3: Sketch the graph of y = log 2 (x).
Rewrite in exponential form: x = 2 y . Create a table of values (choosing values for y): | y | -3 | -2 | -1 | 0 | 1 | 2 | 3 | | :---- | :----- | :----- | :----- | :-: | :-: | :-: | :-: | | x = 2 y | 1/8 = 0.125 | 1/4 = 0.25 | 1/2 = 0.5 | 1 | 2 | 4 | 8 | Plot the points and draw a smooth curve. The graph has a vertical asymptote at x =
0. Example 4: Sketch the graph of y = log 1/2 (x).
Rewrite in exponential form: x = (1/2) y Create a table of values (choosing values for y): | y | -3 | -2 | -1 | 0 | 1 | 2 | 3 | | :---- | :-- | :-- | :-- | :-: | :----- | :----- | :----- | | x = (1/2) y | 8 | 4 | 2 | 1 | 1/2 = 0.5 | 1/4 = 0.25 | 1/8 = 0.125 | Plot the points and draw a smooth curve. The graph has a vertical asymptote at x = 0. 2.3 Laws of Logarithms: These laws are crucial for simplifying expressions and solving logarithmic equations.
Product Rule: log a (xy) = log a (x) + log a (y)
Quotient Rule: log a (x/ y) = log a (x) - log a (y)
Power Rule: log a (x n ) = nlog a (x)
Change of Base Formula: log b (x) = log a (x) / log a (b) log a (a) = 1 log a (1) = 0 Example 5: Simplify log 2 (8) + log 2 (4).
Using the product rule: log 2 (8) + log 2 (4) = log 2 (8 * 4) = log 2 (32). Since 2 5 = 32, log 2 (32) =
5. Example 6: Simplify log 3 (27/9).
Using the quotient rule: log 3 (27/9) = log 3 (27) - log 3 (9). Since 3 3 = 27 and 3 2 = 9, log 3 (27) - log 3 (9) = 3 - 2 =
1. Example 7: Simplify 2log 5 (25).
Using the power rule: 2log 5 (25) = log 5 (25 2 ) = log 5 (625). Since 5 4 = 625, log 5 (625) =
4. Alternatively, 2log 5 (25) = 2 * 2 = 4 since log 5 (25) = 2. 2.4 Solving Exponential Equations using Logarithms: When the bases are not the same, logarithms are essential for solving exponential equations.
Example 8: Solve for x: 3 x =
1
5. Take the logarithm of both sides (any base can be used, but base 10 or e is common on calculators): log(3 x ) = log(15).
Apply the power rule: xlog(3) = log(15).
Isolate x: x = log(15) / log(3).
Use a calculator: x ≈ 2.
4
6
5. Example 9: Solve for x: 5 2x-1 =
2
0. Take the logarithm of both sides: log(5 2x-1 ) = log(20).
Apply the power rule: (2x - 1)log(5) = log(20).
Expand: 2xlog(5) - log(5) = log(20).
Isolate the x term: 2xlog(5) = log(20) + log(5).
Simplify: 2xlog(5) = log(100).
Solve for x: x = log(100) / (2log(5)) = 2 / (2log(5)) = 1/log(5) ≈ 1.431. 2.5 Relationship Between Exponential and Logarithmic Functions: Exponential and logarithmic functions with the same base are inverse functions of each other. This means that if f(x) = a x , then f -1 (x) = log a (x). The graph of a logarithmic function is a reflection of the graph of its corresponding exponential function across the line y = x. Guided Practice (With Solutions)
Question 1: Express the equation 4 3 = 64 in logarithmic form.
Solution: Using the definition of logarithms, log 4 (64) =
3. Commentary: This question directly tests the understanding of converting between exponential and logarithmic forms. The base of the exponent becomes the base of the logarithm, the result of the exponentiation becomes the argument of the logarithm, and the exponent becomes the value of the logarithm.* Question 2: Solve for x: log 2 (x) =
5. Solution: Convert to exponential form: 2 5 = x.
Therefore, x = 32.