Pre- Calculus Mathematics 12 4.1 Exponents Part 1 Goal: Simplify and solve exponential expressions and equations Logarithms involve the study of exponents so is it vital to know all the exponent laws. Review of Exponent Laws x = x x = = x = = x = x y = x = Note: x ± y Simplifying Exponential Expressions In order to simplify any exponential expression, we must first identify a common base in the expression and then use our rules for exponents as necessary. Example 1: Simplify 4 8 2x 3 x Example 2: Simplify ( ) ( ) "
Pre- Calculus Mathematics 12 4.1 Exponents Part 1 Example 3: Simplify " Solving Exponential Equations To solve an exponential equation, use the same principles of simplifying expressions to get a common base on either side of the equation. If the bases on either side are equivalent, then the exponents must also be equivalent. This allows us to use x = x m = n, when x 1, 0, 1 Example 4: Solve 4 = 8 Example 5: Solve " = 81 Practice: Attached Sheet and Page 159 #1(a,b,c,e), 2(a,b,e,f)
Pre- Calculus Mathematics 12 4.1 Exponents Part 1 4.1 Solving Exponential Equations Practice 1. 4 4 = 64 2. 6 6 = "# 3. 2 " = 32 4. 2 2 = 2 5. 64 16 = 16 6. " = 3 "# 7. 81 9 = 27 8. 9 9 = 27 9. 216 = "# 10. 243 9 = 9 Solutions 1. - 3 2. 1 3. 10 4. 0 5. 7 12 6. 4 17 7. 3 4 8. 3 4 9. 1 6 10. 2 3
Pre- Calculus Mathematics 12 4.1 Exponents Part 2 Goal: Explore the basic properties of an exponential function Exponential Functions An exponential function is any function that has a variable in the exponent and a positive base not equal to zero. y = b x where b > 0, b 1 Exploration: Use a table of values to graph the following functions and determine the basic properties of any exponential function y = 2, base b > 1 x y 1 y 2 y = = 2 base 0 < b < 1 All pass through the point: Horizontal Asymptote: Domain: Range: Example 1: Graph the following. State the domain, range, intercept(s) and asymptotes. y = 2 + 1
Pre- Calculus Mathematics 12 4.1 Exponents Part 2 Example 2: Graph the following. State the domain, range, intercept(s) and asymptotes. y = 1 2 1 Practice: Page 160 # 5, 6, 7 We have explored the exponential function; now let s look at the inverse of the exponential function. Exploration: Graph f(x) and its inverse, then algebraically solve for the inverse of the function. f(x) = 2 So if we generalize this for any base y = b and its inverse is x = b Key Points: y = b x = b
Pre- Calculus Mathematics 12 4.2 Logarithms Goals: 1. Define a logarithmic function explore its basic properties 2. Graph a logarithmic function, transform and determine the domain of the function The inverse of an exponential function is another function called a logarithm. Exponential form Logarithmic form Greek: logos word/speech/logic arithmes numbers b y = x log b x = y Note: b > 0, b 1 Formal Definition: A logarithm of a number is the exponent (y) to which a fixed value (b) must be raised in order to get some other number (x) Example 1: Change from exponential to logarithmic form i) 3 3 = 27 ii) 10 4 = 10000 Example 2: Determine the numerical value i) log 4 64 ii) f(x) = log 1/3 27 iii) log 6 x = 3 iv) log 7 (x+2) = 3 Practice: Page 167 #1-4
Pre- Calculus Mathematics 12 4.2 Logarithms Logarithmic Graphs: The graph of an exponential function can be used to determine the graph of a logarithmic function and its basic properties. When b > 0 y = 10 and x = 10 y = log " x Key Points: When 0 < b < 1 y = and x = Key Points: y = log x Logarithmic Domains Since the inverse of an exponential function is a logarithmic function, the domain of a logarithmic function is the range of the corresponding exponential function. y = log x x x > 0, x R}. Remember, the base is: b > 0, b 1 Example 3: Determine the domain of the following: i) y = log ( x 2) ii) y = log ( x 1)
Pre- Calculus Mathematics 12 4.2 Logarithms Transformations of Logarithms A logarithm, like any other function, can be transformed using the principles associated with transforming a function. The transformation will always be in relation to the basic graph, y = log x. Example 4: Sketch each function. i) y = log ( x 2) ii) y = log ( 2 x) + 1 Practice: Page 168 # 5-10
Pre- Calculus Mathematics 12 4.3 Properties of Logarithms Goals: 1. Analyze the properties of logarithmic functions to determine the rules for logarithmic functions 2. Use the rules for logarithms to simplify expressions A logarithm is just the inverse of an exponential function. Just like there are established and proven rules for exponents, there are established and provable rules for logarithms. Rules for Logarithms (The Log Laws) Note: b > 0, b 1 log 1 = 0 log b = 1 Product Rule: log xy = log x + log y Example: Simplify log 5 + log 7 Quotient Rule: log = log x log y Example: Simplify log 24 log 8 Power Rule: log x = n log x Example: Simplify log 16
Pre- Calculus Mathematics 12 4.3 Properties of Logarithms Change of Base: log a = "# "# Example: Find log 7 to 3 decimal places Example 1: Write log " in terms Example 2: Find the exact value of of log 3 and log 5 log 27 3 Example 3: Find the exact value of log 64 Example 4: Find the exact value of log "
Pre- Calculus Mathematics 12 4.3 Properties of Logarithms Example 5: Expand the following log x y z Practice: Page 177 # 1-3 Simplifying Logarithmic Functions: 1. Understand rules #1-6 above. 2. Do not make up your own rules for logarithms. Common mistakes: log A + B log a log b log x 3. Know how to change from exponential form to logarithmic form, and vice versa. y = log x b = x 4. Look for exponential/power relationships between b and x in log x. Example 6: Simplify 3log x 2log 3 Example 7: Simplify 7 2log 7 5
Pre- Calculus Mathematics 12 4.3 Properties of Logarithms Example 8: Simplify log x 4 + log x 5 log x 7 log x 4 Example 9: Simplify log 9x log y Example 10: Simplify "# "# Example 11: Simplify log 10 log 45 log 5 Practice: Page 178 # 4, 5
Pre- Calculus Mathematics 12 4.4 Exponential & Logarithmic Functions Goal: Solve logarithmic or exponential equations Using the rules we have established for logarithmic functions, we can now start to solve logarithmic equations. Remember, an equation has an equal sign, so the solution must equal something. Be careful of logarithmic equations though, the solutions must be a part of the domain of that function y = log x Note: b > 0, b 1 Steps for solving logarithmic functions: 1. If a constant exists in the equation, bring all the logs to one side, constant(s) on the other. OR If no constant exists in the equation, combine logs on each side into single logs with a common base. 2. Convert to exponential form using log x = y b = x OR Use rule: log x = log y x = y 3. Solve the resulting equation for the unknown variable. 4. Reject any extraneous root(s) using: y = log x ; b > 0, b 1 and x > 0 Example 1: Solve for x: log x 5 = 1 log x + 3
Pre- Calculus Mathematics 12 4.4 Exponential & Logarithmic Functions Example 2: Solve for x: log x 2 + log 10 = log x + 3x 10 Practice: Page 183 #1 Expressing equations in terms of stated or defined variables Sometimes we have a question that only contains variables and we need to solve the logarithmic equation in terms of the stated variables. This is just a slight variation on what we have already looked at, just with no numbers Example 3: Solve for A in terms of B and C: 2log A 3 log B = 2 log C Example 3: If = log 5 and b = log 3, what is log 45 in terms of a and b?
Pre- Calculus Mathematics 12 4.4 Exponential & Logarithmic Functions Remember from the start of this chapter, we were able to solve exponential equations with a common base. 3 = 3 " Now, with logarithms, we can solve an exponential equation that doesn t have a common base. 5 = 2 3 " Steps for solving Exponential Equations Variable exponents and no common bases 1. Simplify if possible then log both sides. (base 10) 2. Use log laws to move exponents in front of each log. 3. Expand out the logarithms; bring logs containing unknowns on one side. 4. Isolate and solve for the unknown (usually by factoring), simplify to a single log if possible. 5. Determine a numerical value of the single log (if possible) Example 4: Solve for x: 5 = 2 3 " Practice: Page 184 # 2-4
Tougher questions. Example 5: Solve for x: 3log x + log x = 5 Pre- Calculus Mathematics 12 4.4 Exponential & Logarithmic Functions Example 6: Solve for x: 3log x 2 log 8 = 5 Practice: Page 184 # 5-7