Exponential and Logarithmic Functions. By Lauren Bae and Yamini Ramadurai

Transcription

1 Exponential and Logarithmic Functions By Lauren Bae and Yamini Ramadurai

2 What is an Exponential Function? An exponential function any function where the variable is now the power, rather than the base. This means that x is the exponent of the function, and the base is an integer. To get any exponential expression, multiple the base by itself the number of times dictated by the exponent (also called the power).

3 Simplifying Exponents Division Rule: When solving problems, it may be best to simplify the exponents to make it easier to solve.

4 Example: y=2 x As you can see, the y values of exponential functions rapidly increase, which you can see in their graphs. All the points in an exponential graph can be created by hand, or by typing the equation into a calculator. To get all the points by hand, take the x value and substitute it into the equation. For instance: 2 0 =1 2 1 =2 2 2 =4 2 3 =8 2 4 = =32 x y Example: 2 3 = = 8

5 Exponential Function Graphs Positive Graph Negative Graph

6 Graphing Exponential Functions A basic positive exponential function (not transformed) has an asymptote of y=0, with end behaviors of: As x approaches positive infinity, y approaches positive infinity, and as x approaches negative infinity, y approaches 0. A basic negative exponential function (not transformed) has an asymptote of y=0, with end behaviors of: As x approaches positive infinity, y negative infinity, and as x approaches negative infinity, y approaches 0. An exponential graph has only a horizontal asymptote (y=0). The domain and range of an untransformed exponential function is: D: (-, ) R: (0, )

7 Transformations of Exponential Graphs Exponential graphs can be transformed like linear graphs, up and down or left and right. The basic form is y=a(b) x-h +k. The variable a is the vertical change, the variable h moves the graph from left to right, and the variable k moves the graph up and down. The asymptote of the graph shifts as well.

8 Example y=2 x As you can see, the graph was shifted down 10 units (k) and to the right by 5 units. Horizontal asymptote: y=-10 Domain: (-, ) Range: [-10, ) *Remember that when looking at a horizontal shift, a positive value is shifted to the left, and a negative value is shifted to the right.

9 Applications of Exponential Functions Examples of real life applications for exponential functions include: -Exponential Growth: y=ab x a=initial amount b=growth factor x=amount of time Appreciation Rate: y=a(1+r) x r: rate in decimal form -Exponential Decay: y=ab x Depreciation Rate: y=a(1-r) x 0<b<1 r: rate in decimal form -Half Life (amount of time it takes for initial quantity to reduce to half): y=a(½) x/h H: half life x/h: Number of half lives x: total time -Compound Interest (annually, semiannually, quarterly, monthly, daily): y=a(1+r/n) nt r: rate of interest per year n: # of times compounded in a year t: # of years a: initial deposit -Compounded Continuously: y=pe rt P: initial amount t: # of year r: rate annually

10 Logarithms Logarithms are the INVERSE of exponential functions. Log is used to represent very large or very small numbers. Exponential and logarithmic functions are related: log b y=x <---> y=b x The basic form of a logarithmic function is log b y=x. Base 10 logarithms are the most common, so the notation is simply log(y)=x. If a base is not explicitly written, it means that it is base 10. Log b x is the exponent to the base b that will give the value x. Change of Base log b (a) = log(a) log(b)

11 Log to Exponential Conversions When converting log to exponential form, follow these steps: 1. Move the base of the log to the left side of the equation so that it is now the base of the exponent. 2. The number that the log equation was equal to in the original equation (on the left side of the logarithmic function) is now the exponent 3. The y value (in the case of the example below 8) is now on the left side of the exponential equation. For example - log 2 8=3 -> log8=2 3 -> 8=2 3 or 2 3 =8

12 Solving Exponential Equations using Log To solve exponential equations, you can use log. In most cases, you would just take the log of both sides. Example: 7.5 x = 42.6 log(7.5 x )= log(42.6) x log(7.5) = log(42.6) x= (log(42.6) / log(7.5)) x = 1.88 Example: 3 ll 2x+5 = 20 ll 2x+5 = 20/3 (2x+5) log(11) = log(20/3) 2x+5 = log 11 (20/3) 2x = log 11 (20/3) -5 x = (log 11 (20/3) -5) 2 x = -2.1 Example: 6 2x+1 = 5 4x-5 (2x+1) log(6) = (4x+5) log(5) ((2x+1) / (4x-5)) = log 6 5 *Cross multiply* 2x+1 = 0.9(4x-5) 2x+! = 3.6x x = 5.5 x-3.44

13 Logarithm Graphs A positive logarithmic function with a base of 10 (not transformed) has an asymptote of x=0, with end behaviors of: As x approaches positive infinity, y approaches positive infinity, and as x approaches negative infinity, y approaches 0. A negative logarithmic function with a base of 10 (not transformed) has an asymptote of x=0, with end behaviors of: As x approaches positive infinity, y negative infinity, and as x approaches negative infinity, y approaches 0. A logarithmic graph has only a vertical asymptote (x=0). The domain and range of an untransformed graph is: D: (0, ) R: (-, )

14 Example The orange line is y=log(x), and the purple line is y=-log(x).

15 How to Manually Graph Log Steps to graphing log: 1. Convert the logarithmic function to an exponential function, getting 5 points (if possible, try to make both x and y integers). 2. Since log is the inverse of exponential functions, switch the x and y of the points for the exponential function. These will be your points for your log graph.

16 Example y=log 2 (x) Exponential form: y=2 x Points for Exponential Graph: x y

17 Example for Graphing Log Graph: y=log 2 (x) x y

18 Transformations of Logarithmic Functions Just like an exponential function, logarithmic functions can also be transformed. The basic form is y= a log b (x-h) + k. The variable a is the vertical change, the variable h moves the graph from left to right, and the variable k moves the graph up and down. *h>0, graph moves left. h<0, graph moved right* The asymptote of the graph shifts as well.

19 Example Graph: y = log 3 (x-1)+2 x y 1 N/A The graph moved right 1 unit, and up by 2 units.

20 Properties of Log There are 3 properties of log: Product Property: log(x y) = log(x) + log(y) Quotient Property: log(x/y) = log(x) - log(y) Power Property: log(x n ) = n(log(x)) These properties are used to expand and condense log.

21 Expanding and Condensing Log Examples Expand: log(xyz) = log(x) + log(y) + log(z) log 7 (5/9) = log 7 (5) - log 7 (9) log 5 (x) 4 = 4 log 5 (x) log 3 (x 3 /y 7 ) = 3 log 3 (x) - 7 log 3 (y) Condense: log 6 (10) + log 6 (x) + log 6 (7) = log 6 (70x) log 4 (10) - log 4 (x) = log 4 (10/x) 3 log 12 (x) = log 12 (x 3 ) 6 log 5 (a) - 9 log 5 (b) = log 5 (a 6 /b 9 )

22 Now you try! Expand - log[2(x+1)] 3 Answer: 3 log(2) + 3 log(x+1) Condense - 5 log 2 (x) + 7 log 2 (y) Answer: log 2 (x 5 y 2 )

23 Solving Log Equations One Log: Conversion (log -> exponent), Change of Base Two Logs: Cancel the Log (All bases must be the same), Condense the Logs No Logs: Add a Log, Similar Bases

24 Conversion Use when a variable is attached to the logarithm Example: log 2 (2x+1) = = 2x =2x+1 2x = 15 x=15/2 Example: log(2x-5) = =2x+5 2x-5=100 2x=105 x = 105/2

25 Change of Base Use when the variable is not attached to the logarithm Reminder: log b (a) = log(a) log(b) Example: log 2 (8) = 3x+3 3 = 3x+3 3x=0 x=0 Example: log 8 (25) = x 2 log(25) / log (8) = x 2 x= 1.71

26 Cancel the log *Can only be used if both sides have the same base, and if there is only one log per side Example: log 4 (3x-1) = log 4 (2x+3) 3x-1 = 2x+3 x=4 Now Try: log(x 2 )=log(-2x+15) x 2 = -2x + 15 x 2 +2x-15=0 (x+5)(x-3)=0 x=3, x=-5

27 Condense the Logs Condense the logs so that there is only one log that appears per side. Then, decide which of the other methods to use to finish solving the equation. Example: 3log 2 (x) + log 2 (5) = 7 log 2 (x 3 ) + log 2 (5) = 7 log 2 (5x 3 ) = = 5x 3 5x 3 = 128 x 3 =128/5 x=2.95

28 Add a Log Use this if you cannot get similar bases Move all other values so that it is just the exponent and base on one side, then convert the equation to exponential form and solve from there. Example: 7 x-3 +5 = 30 7 x-3 = 25 log 7 (25) = x-3 x= 4.65

29 Similar Bases Break each base down so that they are the same, cancel the bases, and work with only the exponents. Example: 25 2x = x = 5 3 4x =3 x= ¾ Example: 4 3x 4 2x = 1,048, x = x = 10 x=2

30 Verify Your Answers Any log equation can have an extraneous solution (when the answer is plugged back into the original equation but it doesn t match). To prevent this, always check your answer(s). Example: log 5 (n 2-20) = log 5 (-n) n 2-20= -n n 2 +n-20 = 0 (n-4)(n+5)= 0 n=-5, n=4 n=4 is extraneous, so the answer is n=-5 *Reminder: the log of a negative number does not exist

31 Natural Log A natural base exponential function is y=e x e is a mathematical constant equal to approximately To get e on a calculator, press [2nd] [ ]. Natural log is a normal log equation with a base of e. The notation for natural log is ln(x)=y.

32 Examples 1) e 0.5 = ) e -8 = ) e 2 = ) 3 ln(5) = ln(5 3 ) = ln(125) 5) ln24 - ln6 = ln(24/6) = ln(4) 6) ln(3) = ) ln(¼) = )ln(0.05) = -3 8) e x/4 +3 = 9 e x/4 = 6 ln(6) = x/4 x=7.17 9) e 3x+1 = e 13 3x+1 = 13 3x=12 x=4 10) ln((x-3)/4)=8 e 8 = (x-3)/4 x = 4e 8 +3 x=

33 Application of Log ph - ph = -log 10 [H + ] [H + ] = 10 -ph H + = positively charged hydrogen ion Richter scale- R= log (A/A 0 ) A the measure of the amplitude of the earthquake wave A 0 the amplitude of the smallest detectable wave (or standard wave) Decibels (most basic form)- N db = log 10 (P2 / P1) N db is the ratio of the two power expressed in decibels P2 is the output power level P1 is the input power level

34 Log and Exponential Functions Relation Exponential functions are the inverse of log, and vice versa. log b y=x <---> y=b x You can algebraically prove this or with a graph. An function and its inverse on a graph is identical, but the inverse is reflected over the line y=x. If you look at the points on an inverse graph, the x and y would be switched.

35 Example: Points y= 10 x y=log(x) x y x y

36 Example: Graph Red- y=10 x Blue- y=log(x) Green- y=x Can you see the relationship between the two graphs and their points?

37 Example: Algebra y = log(x) Switch x and y x =log(y) Solve for y y = 10 x f -1 (log(x)) = 10 x y = 10 x Switch x and y x = 10 y Solve for y log(x) = log(10 y ) log(x) = y (log(10)) log(x) = y y= log(x) f -1 (10 x ) = log(x) *Reminder: log 10 (10) is equal to 1. If the base (b) and y are equal, then the log expression is equal to one. *Reminder: To find an inverse, switch x and y, then solve for y

38 Solving Log Equations: Examples 1) log(20) - log(x) = log(5) log(20/x) = log(5) 20/x = 5 5x=20 x=4 2) ln(3x-8) = 2 e 2 = 3x-8 3x = e 2 +8 x = (e 2 +8) / 3 x= ) log(x) - log(4) = -1 log(x/4) = = x/4 1/10 = x/4 10x = 4 x=2/5

39 Try this last one! Equation log(x+21) +log(x) =2 Answer log(x(x+21)) = = x x x 2 +21x-100=0 (x+25)(x-4)=0 x=-25, x=4 x=-25 is extraneous x=4