8: Exponential and Logarithmic Functions 8-1: Exploring Exponential Models 1
A power function raises a variable to a fixed exponent: x 2 or y 3 . Polynomials are just sums of power functions. Radicals are also power functions. An exponential function raises a fixed number (the base) to a variable exponent: 2 x or y
1 . It is generally required that the base be a positive number—you run into problems if you 3 allow non-positive numbers. If the base is greater than 1, then the graph of an exponential function will rise as you look from left to right—this is often called (or models) growth. If the base is between 0 and 1, then the graph will fall as you look from left to right—this is often called decay. An exponential model looks like this: y = a ⋅ n x . Note that this is not the same as y = ( an ) ! x
The graph of this exponential model will pass through ( 0, a ) and (1, an ) . The rising side of the graph will keep rising faster and faster—on up to infinity. The falling side will approach (but never reach!) zero. The number to which the graph falls (but never touches) is called an asymptote. This is a very important idea in Calculus! If you are given some data, and asked to find an exponential model, you may be able to do it algebraically (see one of the following examples). You can also use your calculator—the exponential regression command. We'll talk more about this in class.
Examples [1.] Graph y = 3x . This graph will pass through the points ( 0,1) , (1,3) and ( 2,9 ) .
[2.] Find an exponential model that passes through ( 2,32 ) and ( 3,16 ) . The first point means that 32 = a ⋅ n 2 . The second point tells us that 16 = a ⋅ n3 ; since 16 1 n3 = n 2 ⋅ n , 16 = a ⋅ n 2 ⋅ n = 32 ⋅ n ⇒ = n = . Put that back into the first equation: 32 2 2
1 1 32 = a ⋅ ⇒ 32 = a ⋅ ⇒ 128 = a . 4 2 x
1 The model is y = 128 . 2
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t
2 [3.] A certain population follows the model P ( t ) = 3 , where t is the number of years 3 since the founding of the population. Is this population experiencing growth or decay? 2 Since the base of is less than one, the model is one of decay. 3
8-2: Properties of Exponential Functions Transformations Like all functions, exponential functions can be transformed. The graph of y = a ⋅ n x −c + d is really just the graph of y = n x with the y-values multiplied by a; shifted c units horizontally; shifted d units vertically. That does seem familiar, doesn't it?
Applications Compound Interest When you put money into a bank account, it often draws interest—the bank pays you money, based on the amount of money that you leave in the bank (some economists say that interest is payment for time…). Interest-bearing accounts will gain interest every so often—typically, once a month. When interest is added to the account, then the interest has been compounded. The amount of interest is a percentage of the amount of money in the account—this is called the interest rate. Interest rates are typically given as an annual rate, which is divided up amongst the interest periods (3% interest does not mean that you get an additional 3% each month!) If P dollars (the principal) are invested in an account that bears interest at the annual rate of r, kt
r compounded k times per year, then the value of the investment after t years is V ( t ) = P 1 + . k Note that this only works if the account is left alone—you don't add any more money (only the bank adds interest).
Half-Life A radioactive substance is one that is atomically unstable—every so often, an atom (or part of an atom) will get knocked loose, and the substance will emit radiation. This radiation lowers the mass of the substance—eventually, there will be nothing left. The rate at which a radioactive substance decays is measured by its half-life: the time required for half of the substance to "disappear." If some amount A of a radioactive substance has a half-life of k time-units, then after t timet
1 k units there will be A of the substance remaining. 2 The time units are generally years, but they can be any valid measurement of time (seconds, days, weeks, etc.). HOLLOMAN’S ALGEBRA 2 HONORS
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Continuous Interest There is one exponential function that is more important than the rest—the natural (exponential) function, y = e x . The number e is a constant, called Euler's Number (Euler rhymes with spoiler). The number had been known for some time (first recorded instance: 1618) before Leonhard Euler (Swiss; 1707-1738) used the symbol e to represent the number. The number is x
1 defined to be the limiting value of y = 1 + (the lowest positive value that the function x cannot achieve). There is a nice write-up about this in your textbook. Read it! Our first application of e will be for compounded interest—what would happen if you compounded interest once per day? What about once per second? Once per millisecond? The value will increase, but there is a limiting value—this leads to the Continuously Compounded Interest Formula: V ( t ) = Pe rt . In this formula, P dollars are compounded at an annual rate of r at every possible moment of time, for t years.
Examples [4.] Graph and label y = −3 ⋅ 2 x . This is a graph of y = 2 x with the y-values multiplied by -3; that will flip the graph upside down (across the x-axis) and stretch it away from the axis.
[5.] Write the equation of the exponential graph shown below.
A change of 1 in the x direction takes half of the y-value; that means that the base is
1 (do the 2
3
1 1 same work as above if you don't believe it!). Thus, 6 = a ⇒ 6 = a ⋅ ⇒ a = 48 . The 8 2 x
1 equation is y = 48 . 2 1 (…note that the equation I actually graphed was y = 6 2 result in the same graph?) HOLLOMAN’S ALGEBRA 2 HONORS
x −3
. Can you figure out why they
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[6.] Technetium-99m has a half-life of 6 hours. How much of 50 mg of this substance remains after 25 hours? t
1 6 The amount of this substance after t hours is 50 ; the amount remaining after 25 hours is 2 25
16 50 ≈ 2.784 mg. 2
8-3: Logarithmic Functions as Inverses We've previously said that an exponential function takes in an exponent and returns a number (the base raised to that exponent). We've also previously discussed inverse functions—and the fact that a function has an inverse if the function passes the horizontal line test. So…have you noticed that exponential functions pass the horizontal line test? If an exponential function takes a power and returns a base raised to that power, then the inverse function must take the base raised to the power and return the power. This is called a logarithmic function. The idea of logarithms is credited to John Napier (Scottish; 1550-1617), who invented/discovered them in order to make certain astronomical calculations easier (the properties of logarithms, in the next section, did this for pre-computer and pre-calculator people). If y = n x , then x = log n ( y ) . This is read "the logarithm, base n, of y." This will require that you think backwards when doing problems by hand! Be sure to look at the examples. Note: logarithms with base 10 are (were) so common that many people simply don't write the base when they mean ten…so, when you see log ( x ) , remember that it probably means
log10 ( x ) . You can graph logarithmic functions—since they are inverses of exponentials, just flip the xand y-coordinates! Where exponential graphs have horizontal asymptotes, logarithmic graphs will have vertical asymptotes. Also, remember that the domain of an exponential graph becomes the range of a logarithmic graph (and vice versa). …and once you know how to graph the basic logarithm, you can transform the graph: y = a ⋅ log n ( x − c ) + d . Each of the pieces of this template do exactly what they've done all along!
Examples [7.] Find log 3 ( 81) . Since 81 = 34 , log 3 ( 81) = 4 . [8.] Find log 2 ( 512 ) . Since 512 = 29 , log 2 ( 512 ) = 9 . [9.] Write 32 = 25 in logarithmic form. HOLLOMAN’S ALGEBRA 2 HONORS
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log 2 ( 32 ) = 5 [10.] Write log 4 ( x ) = 3 in exponential form. 43 = x [11.] Graph y = log 4 ( x ) . This will pass through (1, 0 ) , ( 4,1) and (16, 2 ) . The domain of this graph is x ≥ 0 .
8-4: Properties of Logarithms These properties can be derived from the rules of exponents… log n ( a ⋅ b ) = log n ( a ) + log n ( b )
a log n = log n ( a ) − log n ( b ) b log n a b = b ⋅ log n ( a )
( )
Examples [12.] Simplify: log 2 ( 9 ) + 5log 2 ( 3) .
( )
(
log 2 ( 9 ) + 5 log 2 ( 3) = log 2 ( 9 ) + log 2 35 = log 2 9 ⋅ 35
)
[13.] Simplify: 6 log 5 ( x ) + log 5 ( y ) .
( )
( )
6 log 5 ( x ) + log 5 ( y ) ⇒ log 5 x 6 + log 5 ( y ) = log 5 x 6 y [14.] Simplify: 2 logb ( x ) − log b ( y ) .
x2 2 log b ( x ) − log b ( y ) = logb x 2 − logb ( y ) = log b y
( )
(
)
[15.] Expand: log11 4 p 3 .
(
)
( )
log11 4 p 3 = log11 ( 4 ) + log11 p 3 = log11 ( 4 ) + 3log11 ( p )
8-5: Exponential and Logarithmic Equations You can solve exponential and logarithmic equations graphically…I'm assuming that this is not difficult, so I'll not dwell on it here.
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Solving these equations algebraically is a bit harder, and needs some attention. Remember: the goal is to isolate the variable! You will probably have to make use of the rules of exponents and the properties of logarithms. In addition, there is one more useful technique—the change of base formula. log n ( a ) For any valid base n, log b ( a ) = . log n ( b ) As was the case with radical equations, make sure that you check your solutions!
Examples [16.] Solve 32 x−1 = 81 . Convert to logarithmic form: log 3 ( 81) = 2 x − 1 ⇒ 4 = 2 x − 1 ⇒ 5 = 2 x ⇒ x =
5 . This does 2
work (I won't show the check here). [17.] Solve log 2 ( 3x − 2 ) = 16 . Convert to exponential form: 216 = 3 x − 2 ⇒ 65536 = 3 x − 2 ⇒ 65534 = 3 x ⇒
65534 = x ≈ 21844.67 3
[18.] Solve log 5 ( x + 2 ) + log5 ( x − 2 ) = 1 . log 5 ( ( x + 2 )( x − 2 ) ) = 1 ⇒ 51 = x 2 − 4 ⇒ 9 = x 2 ⇒ x = ±3 . Check to see that x = −3 doesn't work—the answer is x = 3 .
8-6: Natural Logarithms For every base, there is a logarithm—however, two bases are more important than the others. First of all, the base of our number system—10. Base 10 logarithms are called common logarithms. They are most commonly used in high schools…(almost) everyone else uses a different kind. In high school, common logarithms are written log ( x ) . The other important base is Euler's number. Base e logarithms are called natural logarithms. This is the only logarithm that is rigorously defined by mathematics! Within high schools, natural logarithms are written ln ( x ) . Outside of high schools, natural logarithms are written log ( x ) . Be careful!
Examples [19.] Simplify: 2 ln (12 ) − ln ( 9 ) .
144 ln 12 2 − ln ( 9 ) = ln 9
( )
[20.] Solve ln ( 2 x − 4 ) = 6 . 3
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e6 = ( 2 x − 4 ) ⇒ e 2 = 2 x − 4 ⇒ e 2 + 4 = 2 x ⇒ 3
e2 + 4 = x ≈ 5.695 2
[21.] Find the exact solution to 4e3 x + 1 = 14 . 13 1 13 13 13 4e3 x = 13 ⇒ e3 x = ⇒ ln e3 x = ln ⇒ 3 x = ln ⇒ x = ln 4 3 4 4 4
( )
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