Scientific Notation Review

Transcription

1 Scientific Notation Review Eamples: Write 5,000 in scientific notation. What power of 10 is.0001? 1) Write 74 in scientific notation. ) What power of 10 is 10,000? ) Write 17 in scientific notation. 4) What power of 10 is 1,000,000? 5) Write.078 in scientific notation. 6) What power of 10 is.001? 7) Write 798,056,000 in scientific notation. 8) What power of 10 is 1? 9) Write.978 in scientific notation. 10) What power of 10 is 10? 11) Write 67 in scientific notation. 1) What power of 10 is ? 1) Write in scientific notation. 14) What power of 10 is.1? ~ 1 ~

2 Lesson #57 Converting between Eponential Form and Logarithmic Form A.A.8 Solve a logarithmic equation by rewriting as an eponential equation A.A.18 Evaluate logarithmic epressions in any base That man is Suzanne s father. That man is the father of Suzanne. These are two different ways to say the same thing. If you were telling a story to your friends you would use different vocabulary than if you were telling your grandmother. In math we also have different ways to say the same thing. Whenever there are two ways to epress the same idea, there are different times where each one is appropriate. Eample 1 - factored vs.simplified: ( )( 5) 15 If I was working with a rational epression such as 15, the factored form would be better so that I could cancel the common factors. If I was working with the equation, ( )( 5) 7, I would prefer the simplified form so that I could set the equation equal to zero. Eample - root vs. fractional eponents: 4 4 If I was evaluation an epression such as If I was working with the equation, 4 8, I would prefer seeing the roots. 4 65, I would prefer the fractional eponent so that I could use the reciprocal eponent to solve the equation more easily. Logarithmic form is just another way to write an equation that is in eponential form. There are different situations where each is desirable, but today we will just be learning how to convert back and forth between the two. The most confusing part of logarithmic form is the fact that there is the word, log, in the mathematical equation. This is just letting you know that you are working in logarithmic form instead of eponential form. ~ ~

3 Let s start with an equation that is an eponential function: y. Like the inverse of a function, when converting to logarithmic form switch and y (the answer and the eponent in this case.) y is the same as or equivalent to log y Notice, the BASE stayed the same () and the and y switched. We just write the base on a lower level (think basement) like this log ywhile everything else is at the normal level. Now, the eponent is the answer. Logs are equal to eponents! The number you are taking the logarithm of, in this case y, is called the argument of the logarithm. How we say it: log y : We do not say: A. Write g log h j in eponential form. B. Write a c b in logarithmic form. Summary Consider the equation 8. log This is equivalent to: Logs bring eponents back down to earth. The is the. The stays the same. ~ ~

4 a. Write in logarithmic form: b. Write in logarithmic form: 16 c. Write in eponential form: log 64 6 d. Write in eponential form: log 7 49 Since logs are equal to eponents, when evaluating a logarithm, you are asking yourself the question, What eponent for the base will give me the argument? Sometimes you can think about this question and determine the answer. At other times, it is helpful to put an where the answer would go and convert the equation to eponential form. Evaluate each of the following logarithmic epressions. (Use PEMDAS when necessary). e. log8 64 = f. 5log 16 = 1 g. log = 9 h. log8 = i. log9 = j. log9 7 = k. log8 16= l. 7log1000 = When there is no base with the log, the implied base is ten. This is known as the common log, which is the log on your calculator. m. log864 log81= 1 log = 15 n. 5 o. log7 1= p. log5 5 6log 7 = log 16 6 ~ 4 ~

5 Solve each of the following equations. (Hint: Convert to eponential form first). q. log 4 r. log 5 s. log 7 t. log 16 u. log 7 v. log ( 50 1 ) log ( ) w log100,000 Summary: Logs are equal to Eponents! 4 16 log 16 4 The logarithm is equal to the original eponent, (4). ~ 5 ~ Note: Many log problems can now be done on the calculator. On all homeworks and the test you must show your work by converting to eponential form! Only use the calculator as a way to check or as a tool to solve the problem when you are stuck.

6 Lesson #58 - Log Functions as Inverses of Ep. Functions A.A.54 Graph logarithmic functions, using the inverse of the related eponential function Write y in logarithmic form. Write log 81 4 in eponential form. Find the inverse of the eponential function, f( ). Steps: 1. If necessary, substitute y for f().. Switch and y (as always).. Solve for y. How: by converting to log form. 4. If necessary, substitute the Notice how the original and the inverse look almost the same ecept for the log. inverse function notation for y. Find the inverse of: f( ) 4 f( ) 1 We can also write the inverse of a logarithmic function by converting the inverse to eponential form. Find the inverse of the logarithmic function, f ( ) log10. Find the inverse of: y log 5 h( ) log.5 ~ 6 ~

7 Graphing logarithms by using the inverse of eponential functions If you do not have the new operating system on your calculator, you need to graph the eponential function first and switch the s and y s of the points to graph the inverse or logarithmic function. Graph f( ) and its inverse,. y= Inverse: y y Note: Instead of using a table, you could also label three points on the graph. What is the domain of f( )? What is the range of f( )? What is the domain of y=log? What is the range of y=log? How are the domain and range of f( ) and y=log related? In what quadrants are parent eponential functions? In what quadrants are parent logarithmic functions? Recall that every eponential function must contain the point. Then every logarithmic function must contain the point. ~ 7 ~

8 1. Graph y=.5 and its inverse. Equation of the Inverse:. Graph y=log 4 and its inverse. Equation of the Inverse: Perform each transformation on the function, f ( ) log. Graph the parent function and its transformations in your calculator to check. Transformation New Equation Domain Range a. Reflection in the -ais b. Reflection in the y-ais c. Vertical Stretch of. d. Horizontal Shift 6 units left. e. Vertical Shift units down ~ 8 ~

9 Summary of eponential and logarithmic parent functions Eponential Function: Domain: All Reals Range: y>0 Common Point: (0,1) y b Inverse (Log Function): y b y log Domain: >0 Range: All Reals Common Point: (1,0) b ~ 9 ~

10 Lesson #59 - Logarithm Rules A.A.19 Apply the properties of logarithms to rewrite logarithmic epressions in equivalent forms Since logarithms are equal to eponents, they have product, quotient, and power rules that are similar to the eponent rules. The Product Rule: The log of a product is equal to the sum of the logs of the factors of the product. The Quotient rule: The log of a quotient is equal to the difference of the logs of the dividend and divisor. The Power Rule: The log of a power is equal to the eponent times the rest of the log. ~ 10 ~

11 Summary of Logarithm Rules Rules Eponents Logarithms y Product Rule log y Quotient Rule Power Rule y log y ( ) y log y The logarithm rules are used for a number of reasons. First we will practice epanding and contracting (simplifying) logarithmic epressions. Note: This is NOT changing a logarithm to eponential form. In order to epand logs, you must write roots in eponential form. Epand the following logarithms. log b y 1.. log c b. log ab c b ac 4. log5 5. log a b 6. y log 4 a 4 7. log ab c 8. log( a b) ~ 11 ~

12 Simplify the following logarithms. This is also called Writing as a single logarithm. I like to call it contracting the logarithm because it is the opposite of epanding. 1) log 4 logbr logbq ) ) logb logb y logb z 4) log 4log y 5) log log y 6) 1 log 1 (log y log z ) Using the logarithm rules to evaluate logarithms Sometimes, when you do not know the value of either logarithm alone, using the rules creates a logarithm that you can evaluate. (Show your work here. Only use the calculator to check.) 1) log1 9 log1 16 ) log410 log45 ) log61 log6 = Just as you need to be comfortable converting between logarithmic and eponential form, you will also want to learn these rules well enough to epand and contract within logarithmic form for the rest of the unit. ~ 1 ~

13 Lesson #60 Common and Natural Logs A.A.8 Solve a logarithmic equation by rewriting as an eponential equation A.A.18 Evaluate logarithmic epressions in any base Common Logs: log() A logarithm can have any positive value as its base, but two log bases are more useful than the others. The base-10, or "common", log is popular for historical reasons, and is usually written as "log()". For instance, ph (the measure of a substance's acidity or alkalinity), decibels (the measure of sound intensity), and the Richter scale (the measure of earthquake intensity) all involve base-10 logs. If a log has no base written, you should assume that the base is 10. ~Purple Math Evaluate the following common logs without a calculator. 1. log log 10. log1,000, log1 5. log log 45 Since common logs are used so often in real life, we have a button for them on the calculator. Evaluate each of the following logarithms to the nearest hundredth. (Problems 1-6 could also have been evaluated on the calculator.) log log60, log log7,098, log 7 log log 1. log log 4 log Why should you epect log15 to be a number between and? ~ 1 ~

14 Solving for the argument Round to the nearest tenth. (This is also known as finding the antilogarithm.) 1. log 14. log log log.5 Natural Logs: ln() The other important log is the "natural", or base-e, log, denoted as "ln()" and usually pronounced as "ell-enn-of-". (Note: That's "ell-enn", not "one-enn" or "eye-enn"!) Just as the number e arises naturally in math and the sciences, so also does the natural log, which is why you need to be familiar with it. ~Purple Math ln log e Evaluate the following natural logs without a calculator. 17. ln e 18. ln1 This is really what ln() means. If you feel more comfortable, you can always rewrite ln() with the word log and the base e. 19. ln e 1 0. ln e Since natural logs are also used so often in real life, we have a button for them on the calculator as well. Evaluate each of the following logarithms to the nearest hundredth. (Problems 17-0 could also have been evaluated on the calculator.) 1. ln15. ln.009 ln 4 ln e. ln ln47 ~ 14 ~

15 Solving for the argument Round to the nearest tenth. 5. ln 6. ln 1 7. ln ln.6789 Finding the domain of a logarithmic function Recall from lesson #59 that the domain of a basic logarithmic function such as y log is 0. You cannot take the log of a negative number or zero. In other words, the argument must be greater than zero. We will use this fact to find the domain and range of other logarithmic functions where the argument is a binomial. Find the domain of each logarithmic function. f ( ) log 9. 4 ( 5) 0. y log 6 1. g( ) ln. h( ) ln( 7). y log 5 ( 4) 4. y log( 4) 5. y log 5 ( 1) ~ 15 ~

16 Lesson #61 Using Natural Logs and the Power Rule to Solve Eponential Equations A.A.19 Apply the properties of logarithms to rewrite logarithmic epressions in equivalent forms A.A.6 Solve an application which results in an eponential function A. What is the base of ln? B. Epand: ln If you are asked to solve an eponential equation but you are unable to find a common base, you are probably epected to use logarithms to solve it. If you take the natural log ( ) of both sides of an equation, and then use the power rule, eponential equations can be turned into linear ones that can be solved much more easily. In the equation 4, you could find a common base. In the equation 5, you could not because 5 is not a power of. You need logarithms to solve it algebraically. TAKING THE NATURAL LOG OF BOTH SIDES IS NOT THE SAME AS CONVERTING TO LOGARITHMIC FORM. Eample: Find to the nearest hundredth: Use SADMEP to isolate the eponential part if necessary.. Take the natural log of both sides and use the power rule to put the eponent in front of the log.. Solve for. 1) Find to the nearest hundredth: ) Find to the nearest tenth: ~ 16 ~

17 ) Solve for t to the nearest tenth: 1t Note: If you get a repeating decimal at some point while you are solving the problem, be sure to copy and paste it to keep the entire answer. Otherwise your final answer might be slightly off. 4) Find to the nearest hundredth: 5(1.06) 150 5) Find t to the nearest tenth: e t Eponential Word Problems Revisited Each of the last problems was set up to be like the ones we found last unit with eponential growth and decay. Last unit we had to use guess and check to solve them. Now you have an algebraic method to do so. This will be faster, give you an accurate answer, and it is the only way you will receive full credit on the regents. 6) A small country whose current population in 010 is 00,000 people, has been eperiencing a 10% population increase every year. In what year will the population reach 1 million people? ~ 17 ~

18 7) The equation, A=P(1+r/n) nt is used for modeling compound interest. A is the final amount, P is the principal, r is the interest rate, n is the number of times the interest is compounded per year, and t is the number of years. If $100 is invested at 8% interest compounded quarterly, after how many years will the amount in the account double? Round to the nearest tenth of a year. 8) The number of dandelions in your lawn is increasing continuously at a rate of 5% per day, and there are 75 dandelions now. After how many weeks will the number of dandelions rt A A e.) reach 00? (The equation for continuous growth is 0 Note: With the new operating system on the calculators, you can also convert many of these equations into logarithmic form to solve them. It is still important that you understand the process of taking the natural log of both sides of the equation. You will be using it in your future math studies. ~ 18 ~

19 Lesson #6 More with Logarithmic Equations A.A.8 Solve a logarithmic equation by rewriting as an eponential equation A.A.19 Apply the properties of logarithms to rewrite logarithmic epressions in equivalent forms Logs on Both Sides now that we know more about how to work with them, we are going to revisit equations with logarithms. The easiest types of logarithmic equations are those with logarithms on both sides. Why? Because you can simply cancel them on both sides log log y y. First you must write both sides as a single logarithm 1) Solve for y: 5log log y ) Solve for : log 9 log log ) Solve for y: ln56 ln ln8 1 4) Solve for : log 7 49 log 7 14 log 7 1 5) Solve for : log5 log5 log51 6) If log k clog v log p, k equals (1) v c c p () v p () ( vp) c (4) cv p 1 6) Epress in terms of a, b, and c: log (log a logb log c) ~ 19 ~

20 Sometimes you will have to work in the reverse order. 7) A black hole is a region in space where objects seem to disappear. A formula used in the GM study of black holes is the Schwarzschild formula, R. Based on the laws of c logarithms, log R can be represented by (1) log G log M log c () log G log M log c () log log G log M log c (4) log GM log c 8) Banks use the formula A P( 1 r) when they compound interest annually. If P represents the amount of money invested and r represents the rate of interest, which epression represents log A, where A represents the amount of money in the account after years? (1) log P log( 1 r) () log P log1 r () log P log( 1 r) (4) log P log log( 1 r) A logarithm on one side of the equation When there is a logarithm on only one side of the equation you can write the equation in eponential form and solve from there. You must first make sure that the logarithm is simplified or written as a single logarithm. From there you can solve the eponential equation using any algebraic method we have learned. Round to the nearest hundredth when necessary. 9) Solve the following equation log4 10) Solve for : log7 0 log 11) 10 log 1) log 6( 1) log 6( 4) 1 ~ 0 ~

21 1) Solve for : 14) Solve for to the nearest ten thousandth: 6ln ) Solve for to the nearest tenth: ln( 4) ln(6) More with evaluations Note: For the problems, write the logarithm as an eponential equation to solve. As always simplify (write each as a single logarithm) first. (I know you can also do these in your calculator, but you must show your work!) 16) Find log4 18 log4 to the nearest hundredth. 17) Find log 1 5 to the nearest hundredth. 18) Find log 16 log to the nearest hundredth. ~ 1 ~