Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Define and sketch graphs of exponential and logarithmic functions.
• Apply the laws of logarithms to simplify expressions.
• Solve exponential and logarithmic equations.
• Model and solve real-world problems involving growth and decay.
• Determine the inverse of exponential and logarithmic functions.

Lesson summary

Exponential and logarithmic functions are fundamental mathematical tools that describe relationships where quantities grow or decay rapidly. These functions are not just abstract mathematical concepts; they are powerful models for understanding real-world phenomena that affect our daily lives in South Africa and globally. From calculating compound interest on savings accounts to modeling the spread of diseases like COVID-19 or understanding the depreciation of a car's value, exponential and logarithmic functions provide the framework for critical decision-making and informed citizenship.

Lesson notes

2.1 Exponential Functions: An exponential function is a function of the form f(x) = a x , where a is a constant called the base, and a > 0 and a ≠

1. The variable x is the exponent.

Base: The base a determines whether the function represents growth or decay. If a > 1, the function represents exponential growth. As x increases, f(x)* increases rapidly. If 0 x always passes through the point (0, 1) because a 0 =

1. It has a horizontal asymptote at y = 0 (the x-axis).

Example 1: Growth - Population Growth Suppose the population of Gauteng is growing at a rate of 2% per year. If the current population is 15 million, we can model the population P after t years using the exponential function: P(t) = 15(1.02) t , where P is in millions.

Example 2: Decay - Depreciation A new car costs R250,000 and depreciates at a rate of 15% per year. The value V of the car after t years can be modeled by the exponential function: V(t) = 250000(0.85) t , where V is in Rand. 2.2 Logarithmic Functions: A logarithmic function is the inverse of an exponential function. If y = a x , then x = log a y. The logarithmic function is written as f(x) = log a x, where a is the base, a > 0, and a ≠

1. The logarithm log a x is the exponent to which a must be raised to obtain x.

Base: The base a is the same as the base of the corresponding exponential function.

Graph: The graph of y = log a x passes through the point (1, 0) because log a 1 =

0. It has a vertical asymptote at x = 0 (the y-axis). The domain is x > 0, and the range is all real numbers.

Example 3: Converting between Exponential and Logarithmic Form: 2 3 = 8 is equivalent to log 2 8 = 3 5 -2 = 1/25 is equivalent to log 5 (1/25) = -2 2.3 Laws of Logarithms: These laws are essential for simplifying logarithmic 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 ) = n log a x Change of Base Rule: log a x = log b x / log b a (Useful for using calculators with only base-10 or base-e logarithms)

Example 4: Simplifying Logarithmic Expressions: Simplify: log 2 8 + log 2 4 - log 2 2 Solution: log 2 8 + log 2 4 - log 2 2 = log 2 (8 4 / 2) (Using Product and Quotient Rules)* = log 2 16 = log 2 (2 4 ) = 4 (Because 2 4 = 16) 2.4 Solving Exponential and Logarithmic Equations: Exponential Equations: To solve exponential equations, we often try to get the same base on both sides of the equation or use logarithms to isolate the variable.

Example 5: Solving an Exponential Equation: Solve for x: 3 x+1 = 27 Solution: 3 x+1 = 3 3 (Express 27 as a power of 3) x + 1 = 3 (If the bases are the same, the exponents must be equal) x = 2 Logarithmic Equations: To solve logarithmic equations, we use the laws of logarithms to condense the expression into a single logarithm and then convert it to exponential form.

Example 6: Solving a Logarithmic Equation: Solve for x: log 2 (x + 2) = 3 Solution: x + 2 = 2 3 (Convert to exponential form) x + 2 = 8 x = 6 Important

Note: Always check for extraneous solutions when solving logarithmic equations, as the argument of a logarithm must be positive. 2.5 Inverse Functions: The inverse of an exponential function y = a x is the logarithmic function y = log a x. The inverse of a logarithmic function y = log a x is the exponential function y = a x .

Example 7: Finding the Inverse Function: Find the inverse of f(x) = 2 x .

Solution: Replace f(x) with y: y = 2 x Swap x and y: x = 2 y Solve for y: y = log 2 x Replace y with f -1 (x): f -1 (x) = log 2 x Guided Practice (With Solutions)

Question 1: Sketch the graph of y = 3 x . State its domain, range, and asymptote.

Solution: Graph: The graph will be an increasing exponential curve passing through (0, 1).

Domain: All real numbers, or (-∞, ∞)

Range: y > 0, or (0, ∞)

Asymptote: y = 0 (the x-axis)

Commentary: This question tests the understanding of the basic exponential function. Understanding the key features like the asymptote is important.

Question 2: Simplify: 2log 5 10 - log 5 4 Solution: 2log 5 10 - log 5 4 = log 5 10 2 - log 5 4 (Power Rule) = log 5 100 - log 5 4 = log 5 (100/4) (Quotient Rule) = log 5 25 = log 5 5 2 = 2

Commentary: This question tests the application of the laws of logarithms. It is crucial to apply the power rule first.

Question 3: Solve for x: 4 x = 8 x-1 Solution: 4 x = 8 x-1 (2 2 ) x = (2 3 ) x-1 (Express both sides with the same base) 2 2x = 2 3(x-1) 2x = 3x - 3 (Equate exponents) x = 3

Commentary: This question requires expressing both sides with the same base before equating the exponents.

Question 4: The population of a certain bacteria colony doubles every 3 hours. If the initial population is 1000, what will the population be after 12 hours?

Solution: Let P(t) be the population after t hours. We have P(t) = 1000 2 (t/3) *, since the population doubles every 3 hours.