Logarithms involve the study of exponents so is it vital to know all the exponent laws.

Transcription

1 Pre-Calculus Mathematics Exponents Part 1 Goal: 1. 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 ( ) ( ) ( ) ( ) Note: ( ) 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 ( ) ( )

2 Pre-Calculus Mathematics Exponents Part 1 Example 3: Simplify ( ) Example 4: 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 Example 5: Solve

3 Pre-Calculus Mathematics Exponents Part 1 Example 6: Solve ( ) ( ) ( ) Example 7: Solve Example 8: Solve Practice: Attached Sheet and Page 159 #1 + 2

4 4.1 Solving Exponential Equations Practice ( ) 10. Solutions

5 Pre-Calculus Mathematics Exponents Part 2 Goals: 1. Explore the basic properties of an exponential function 2. Apply exponential functions to solve compound interest problems 3. Apply exponential functions to solve growth and decay problems Exponential Functions An exponential function is any function that has a variable in the exponent and a positive base not equal to zero. where b > 0, b 1 Exploration: Use your graphing calculator to graph the following functions and determine the basic properties of any exponential function,, ( ) base 0 < b < 1 All pass through the point: Horizontal Asymptote: Domain: Range: Example 1: Without technology, graph the following. State the domain, range, intercept(s) and asymptotes.

6 Pre-Calculus Mathematics Exponents Part 2 Example 2: Without technology, graph the following. State the domain, range, intercept(s) and asymptotes. ( ) Practice: Page 160 # 5, 6, 7 Applications of Exponential Functions There are a variety of situations where exponential functions can be applied to solve real life problems. Though these may seem like unique situations, they are all just variations of the basic exponential Compound Interest: Interest calculated on principle invested and then added to the original investment ( ) where A = P = r = n = t = Example 3: Find the interest earned if $2500 is invested at 8.5%/a compounded semi-annually for 4 years

7 Example 4: After 10 years, who has more money? Miley: $5000 invested at 8.5%/a compounded semi-annually Liam: $7000 invested at 6.25%/a compounded annually Pre-Calculus Mathematics Exponents Part 2 Earthquakes/pH The Richter scale and a ph scale are just power of 10 scales, meaning in, the base is 10. Example 5: How many times more powerful is an earthquake measuring 8.4 on the Richter scale compared to an earthquake measuring 4.2? Growth and Decay Discrete Growth/Decay growth/decay occurs at a specified rate over a specific time period. 7.5 growth r =, 15% depreciation r =, population doubles r = population triples r =, radioactive half-life decay r = ( ) F = Final amount I = initial amount r = rate of growth/decay t = total time p = period of growth/decay

8 Pre-Calculus Mathematics Exponents Part 2 Example 6: A photocopier is set to reduce an image by 8%. What is the copy size after 4 reductions? Continuous Growth/Decay growth/decay occurs in a continuous time period: Eg: bacteria and plant growth, human population growth, radiation absorption and decay F = Final amount I = initial amount r = continuous rate of growth/decay per unit time t = total time Example 7: A bacterial population of 3000 doubles every 40 min. How many bacterial exist after 4 hours. Practice: Page 162 # 9

9 Pre-Calculus Mathematics Logarithms Goals: 1. Define a logarithmic function, explore its basic properties and transform the function 2. Change a function from logarithmic to exponential form and vice versa and solve for unknowns 3. Find the inverse of a given logarithm or exponential function 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. So if we generalize this for any base and its inverse is Key Points: (, ) (, ) (, ) (, ) (, ) (, ) The inverse of an exponential function is another function called a logarithm. Exponential form Logarithmic form Greek: logos word/speech/logic arithmes numbers Note: 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)

10 Example 1: Change from exponential to logarithmic form Pre-Calculus Mathematics Logarithms i) 3 3 = 27 ii) 10 4 = 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 Logarithmic Graphs: The graph of an exponential function can be used to determine the graph of a logarithmic function and its basic properties. When Key Points: (, ) (, ) (, ) (, ) (, ) (, )

11 Pre-Calculus Mathematics Logarithms When ( ) ( ) Key Points: (, ) (, ) (, ) (, ) (, ) (, ) 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. {. Remember, the base is: Example 3: Determine the domain of the following: i) ii) 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, Example 4: Sketch each function. i)

12 Pre-Calculus Mathematics Logarithms ii) Inverse Log Functions To find the inverse of a log function, always refer to the relationship between logarithms and exponential functions. Steps to finding inverse: 1. Reverse 2. Isolate the power or the logarithm 3. Switch: exponential form log form 4. Solve for y Example 5: Determine the inverse of the following i) ii) Practice: Page 168 # 5-10, 13

13 Pre-Calculus Mathematics 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: Product Rule: Example: Simplify Quotient Rule: Example: Simplify Power Rule: Example: Simplify

14 Pre-Calculus Mathematics Properties of Logarithms Change of Base: Example: Find to 3 decimal places Example 1: Write in terms Example 2: Find the exact value of of log 3 and log 5 Example 3: Find the exact value of Example 4: Find the exact value of

15 Pre-Calculus Mathematics Properties of Logarithms Example 5: Expand the following 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: ( ) ( ) 3. Know how to change from exponential form to logarithmic form, and vice versa. 4. Look for exponential/power relationships between b and x in. Example 6: Simplify Example 7: Simplify

16 Pre-Calculus Mathematics Properties of Logarithms Example 8: Simplify Example 9: Simplify Example 10: Simplify Example 11: Simplify ( )( ) Practice: Page 178 # 4, 5

17 Pre-Calculus Mathematics Exponential & Logarithmic Functions Goal: 1. 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 Note: 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 OR Use rule: 3. Solve the resulting equation for the unknown variable. 4. Reject any extraneous root(s) using: ; Example 1: Solve for x: ( ) ( )

18 Pre-Calculus Mathematics Exponential & Logarithmic Functions Example 2: Solve for x: ( ) ( ) 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: Example 3: If, what is in terms of a and b?

19 Pre-Calculus Mathematics Exponential & Logarithmic Functions Remember from the start of this chapter, we were able to solve exponential equations with a common base. Now, with logarithms, we can solve an exponential equation that doesn t have a common base. 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: Practice: Page 184 # 2-4

20 Tougher questions. Example 5: Solve for x: Pre-Calculus Mathematics Exponential & Logarithmic Functions Example 6: Solve for x: Practice: Page 184 # 5-7

21 Pre-Calculus Mathematics 12 The Natural Logarithm Goal: 1. Solve logarithmic equations with natural logarithms Common Logs vs. Natural Logs Example 1: Solve 4 2 x 20 using common logs. Example 2: Solve 4 2 x 20 using natural logs. When solving an exponential equation, it does not matter whether it is with common logs or natural logs. But if the exponential equation has a base of e, always use natural logs. 2x Example 2: Solve e 8 Practice: See back page

22 Pre-Calculus Mathematics 12 The Natural Logarithm Solve each equation using common logarithms. x 1.) 8 10 x 2.) ) 1.8 x Solve each equation using natural logarithms, 4.) 3 5 x 85 5.) 4 2 x 25 x 6.) ) 7 x 3 4 x Solve each equation 8.) e 0. 04t 9.) 12 e x 10.) e t Solutions: 1) x = ) x = ) x = ) x = ) x = ) x = ) x = ) t = ) x = ) t =

23 Pre-Calculus Mathematics Applications of Exponential & Logarithmic Functions Goal: 1. Apply logarithmic or exponential functions to solve real world problems Using the rules we have established for exponential and logarithmic functions, we can now start to apply these to real life situations. Recall from 4.1: Compound Interest Discrete Growth Continuous Growth ( ) ( ) Example 1: Danny-Boy inherits $ and invests it in a guaranteed investment certificate (GIC) at 6 %. How long will it take to be worth $ it is compounded a) monthly b) continuously? Example 2: A major earthquake of magnitude 8.3 is 120 times as intense as a minor earthquake. Determine the magnitude, to the nearest tenth, of the minor earthquake.

24 Pre-Calculus Mathematics Applications of Exponential & Logarithmic Functions Example 3: The half-live of carbon-14 is 5730 years. A bone sample is found to have 49.5% of the C-14 remaining. Determine the age of the bone. Example 4: The population of a fish in a lake is increasing at a rate of 3.5% per year. How long will it take the fish population to double? Practice: Page 192 # 1-11