Merced College; Don Power

 

INTERMEDIATE ALGEBRA - CH 9, LECTURE

 

 

9.1  Exponential Functions   

 

Intro:  find function for the max height of a bouncing ball on each bounce, if it falls initially from 1 m, and each subsequent bounce is 2/3 the previous height.

      For a table of values:

            Bounce numbers -- arithmetic sequence

            Heights -- geometric sequence

 

Def:  exponential function:  one of the form f(x) = bx, b>0, b¹1.

      Note:  the variable is [in] the exponent

 

Ex:  Find f(3) if f(x)=(1/2)x

 

Ex:  if half-life if I-131 is 8 days

      Amt left in bloodstream after successive 8-day periods is 1/2, 1/22, 1/23, etc of original amt,

      i.e. 1/2n, where n counts 8-day periods.

Convert n to t/8 to count in days.

      So formula is A(t) = A0 / 2t/8, or more standard form A(t) = A0 * 2-t/8

      Handle this as a variation problem to find the constant Ao:  it is always the initial amount (why?)

      Ex 2 asks to find the amount left in the bloodstream after

            a.  10 days:  apply formula

            b.  16 days:  Easier: recognize that this is 2 increments of the half-life: halve initial amount twice

 

Ex:  Table and graph of exponential function y = 2x

      With x positive, graph increases rapidly

      With x negative, graph approaches the y-axis -- why can't this function be negative? (or zero?)

            x-axis is a "horizontal asymptote"

      Two key ordered pairs to learn:

            (0,1) and (1,base)

 

Ex:  Table and graph of y=(1/3)x   --   Note: same as y = 3-x

      Graph with (0,1) and (1,base) and horizontal asymptote on x-axis:  this is a decreasing function

 

Compound Interest Formula is exponential:

      A(t) = P (1 + r/n)nt

      Defs of letters P, r, n, t, A

 

Ex:  Deposit $500 at 8% compounded monthly for 5 years

      How to get accurate result on calculator, since 8/12 does not divide evenly

      Compare result with text's example (compounded quarterly)

 

Continuous Compound Interest:

      A(t) = Pert

 

Ex:  Same problem:  $500 at 8% for 5 years

      Compare result with quarterly and monthly results

 

Second question:  how long to grow to $1000? -- That's what we need this chapter for.

 

9.2  The Inverse of a Function

 

Def:  A relation is a set of ordered pairs;  a function is a relation for which each first coordinate (usually x) is associated with a unique second coordinate (usually y)

 

The inverse of a relation (or function) is a new relation obtained by reversing the order of the coordinates in each ordered pair. -- interchange the x's and y's.

 

Ex:  inverse of f = {(–1,1),(0,0),(1,1),(2,4),(3,9)} is {(1,–1), (0,0),(1,1),(4,2),(9,3)}

 

Is the original set a function?  Is the inverse a function?  What could be done to make the inverse a function?

 

In the special case where both the original relation and the inverse are functions:

      a.  If the original relation is a function named f, then the inverse can be named f-1 (read "f inverse")

            Inverse of the function y=f(x) is denoted y=f-1(x)

b.  Fact:  the inverse function reverses the effect of the original function.  (Illustrate with set above)

c.  We say the function is "one-to-one" -- each x is assoc with exactly one y and vice versa.

 

Effect on domains and ranges:  interchanged.  Domain of relation is same a range of inverse, and vice versa

 

Effect on the graph of the function:  all points (hence entire graph) reflect across the diagonal line y=x.

      Illustrate with the two sets

 

Finding the inverse of a function written as a formula:

      1.  Write as y=

2.  Interchange the x's and y's

      3.  Solve for the new y.

      4.  Write in standard notation for inverses, using the appropriate name for the function (f-1, g-1, etc.)

 

      Ex:  Graph f(x) = x2.  Is it 1-1?  Will the inverse be a function?  What can we do to make it 1-1?

            Find inverse of the restricted function g(x)=x2, x≥0, and graph results;

Contrast this with the graph of the inverse of the complete function  Graph f(x) = x2, note inverse of this one is not a function.

 

      Ex:  Find the inverse of f(x)=(x-4) / (x+2)    Ans:  f-1(x)=(2x+4) / (1-x)

            Check points in f:  (0,-2),(-1,-5),(4,0)

 

      Ex:  Graph the function and its inverse: f(x) = 2x
                  Can we find a formula for the inverse function?  Not yet -- we can't yet solve for y in x=2y

                  We'll need the next lesson to be able to write this as a formula, solved for y = func of x

 

9.3  Logarithms are Exponents

 

Motivation: 

      Review:

            In 7.1, we discussed exponential functions y = bx, also known as y = expb(x)

                  We also graphed them: 

                        Pass through (0,1) and (1,base),  [third point:  (–1, 1/base)]

                        On one side, they increase very rapidly; and on the other, they approach x-axis

            In 7.2, we showed how to find the inverse of a one-to-one function

                  We know it's one-to-one if it passes a horizontal line test

                  We swap the letters x & y wherever we see them

                  Then we solve for the new y

 

Now:  Put these together

                             

      Task:  Find the inverse of y = bx 

                  First, it's one-to-one, by horizontal line test.

                  We know we need to swap x & y, then solve for the new y

                  Swapping, we get x = by 

                  Solving is a problem:  we've never solved equations where the variable is in the exponent

 

Basic task of lesson 7.3 is to solve by=x

      Nothing we have seen so far in algebra will give us a solution

      So we resort to some fakery:

            Instead of solving the equation, we simply describe what y does in the equation:

 

                  "y is the exponent of the base b that gives us x"

 

Being mathematicians, we'd like to shorten this, using symbols,

            What could be more natural than y = expb(x)

            Problem:  we've already used the function "exp" for the exponential function

 

So we have to invent a name for this inverse function, and the name that was picked was "logarithm", or "log"

 

Def:  y=logbx is equivalent to x=by

      "y is the log, base b, of x"

      Interpretation:  "y is the exponent of the base b that gives us x (a log is an exponent)

 

Conversions

 

From log form to exponent form:

            1.  Identify the base

            2.  Identify the exponent  (use "log = exponent" - look on other side of equals sign)

            3.  Fill in third number (the "power" - example: the integer powers of 2 are 1, 2, 4, 8, 16, 32, ...)

 

      From exponent form to log form

            Same steps:  identify base, then exponent, then 3rd number (the power)

 

      Examples:

            1.  53=125 à log                           Ans:  log5125=3

            2.  log1010000 = 4 à exp              Ans:  104=10000

            3.  6-2=1/36 à log                   Ans:  log6(1/36) = -2

            4.  log3(1/81) = -4 àexp               Ans:  3-4 = 1/81

            5.  rs=t à log                                 Ans:  logrt = s

            6.  logde=f à exp                           Ans:  df=e

            7.   3x–5 = 7 à log                          Ans:  x–5 = log37 à x = 5 + log37

            8.   log2(t+7) = 3 à exp                 Ans:  23 = t + 7 à 8 = t + 7 à 1 = t

 

 

Graphing log functions y=logbx

      As inverse of an exponential function

      Features:
            Two important points:  (1,0), (base,1)

            Asymptote:  x=0  (that is, the y-axis)

 

Special identities:

      logb1=0

      logbb=1

      logbbx=x           and               (Convert each to the alternate form)

 

Application:      Richter scale:  Richter reading is M = log10T, where

      M = magnitude of quake on Richter scale

T = size of shock wave

(more precisely, ratio between earth movement of this quake to earth movement of a "standard" quake whose amplitude is 1 micron = 1´10-6 m; energy combines earth movement and time)         Note the 1989 Loma Prieta quake had magnitude 7.1;

Per the USGS web site,

The Alaskan earthquake that is outstanding in the memory of most occurred in the Anchorage area on March 27, 1964 (March 28, 1964 UTC). The magnitude 8.5 [recalculated to 9.2] shock devastated downtown Anchorage and left homes twisted and broken in the residential section of Turnagain. A tsunami virtually destroyed many of Alaska's coastal towns and spread death and destruction along the west coast of the United States, Hawaii, and Canada.

Since the temblor occurred on Good Friday, a holiday for schools, and at a time when most people were out of office buildings and on their way home from work, few deaths were caused by the earthquake itself. But 122 persons were drowned by the ensuing tsunami waves: 107 in Alaska, 11 in California, and 4 in Oregon.

Compare the Alaska quake to the Loma Prieta quake, i.e. find TA / TLP

 

 

9.4  Properties of Logarithms

 

Rules:

      1.  log of a product = sum of logarithms:  logb(xy) = logbx + logby

            compare exponent rules:  what do you do with exponents when you multiply?

      2.  similarly, log of a quotient = difference of logarithms

            compare exponent rules:  what do you do with exponents when you multiply?

      3.  log of an expression with an exponent = product of the exponent and the log:  logbxr = r*logbx)

            compare exponent rules:  what do you do with exponents when you raise a power to a power?

 

Proof of first rule:  let f= logbx, g=logby, solve for x and y, calculate logb(xy).

Alternate demonstration:     From laws of exponents:  23 * 25 = 28

                                                Let these be called A=23, B=25, then AB=28

                                                "exp of 2 that gives A + exp of 2 that gives B = exp of 2 that gives AB"

 

Tasks

 

      Expand, as much as possible,  a log expression involving products, quotients, exponents

            Application:  needed in apps where we need to deal with simpler terms (e.g. in calculus)

 

Ex:  X8, 10, 12, 22, 26

 

      Write as a single logarithm, a expression containing sums, differences, multiples of logs.

            1.  Coefficients [of log terms] become exponents [of the arguments]

            2.  Positive log terms:  arguments become factors in numerator;

                  Negative log terms:  arguments become factors in denominator

 

            Ex:  X30, 38, 42

 

Solving log equations (X43-64)  Steps:

      1.  Get all log terms together on one side of the equation

      2.  Combine into a single log expression. 

            Result must be of form logb(expr)=C -- coeff of the log term must be 1

      3.  Convert to exponential form

      4.  Solve for the variable

      5.  You must check to ensure that the solutions do not make the argument of a log expression £ 0.

 

      Ex:  X46, 50, 56, 60

      Ex:  Solve  log6(x+1)+log6(3x-2)=log6x+1

 

Common Logarithms and Natural Logarithms

 

Common log:  log with a base of 10 (b=10).   Notation:  log10x = log x

      So:  rewrite Richter formula as M = logT

      Note log(1000)=3, log(1/1000)= -3,  etc.

      Calculator use to find log x if x is given

      Calculator use to find x if log x = given number (i.e. finding antilogarithms, antilogs)

 

Natural logs

      Use to simplify expressions (60,65)

      Solve some equations (45-46)

 

9.5  Exponential Equations and Change of Base    

 

Useful tool:  Change of base theorem (makes it possible to use calculator to evaluate logs for any base)

      Forms:   for conversion to a base 10 log,

                         for conversion to a natural log (base e)

                        most general:   -- not applicable to a calculator

      Ex:  Evaluate log6(15) with a calculator.  Result:  approx 1.5114.  Check:  6 1.5114 ≈ 15

 

Special case -- both sides of the equation can be written as exponential expressions with the same base:

      If the bases are the same, then the exponents must be the same

 

      Ex:       47x = 45

165 = 83x -- write both 16 and 8 as powers of 2

 

More general:

      1.  Get equation into form baseexponent = 3rd expression.

Note --  coeff. of the exponential expression must be 1

      2.  Take logs (or natural logs) of both sides.  

[if the base is 10 or e, choose the corresponding log form]

      3.  The exponent of the log argument becomes the coefficient of the log expression.

      4.  Solve for the variable

 

      Ex:       Solve  2*85-3x = 10

                  Solve  4e3x+1 = 3

Alternate procedure:

      1.  (same)   Get equation into form baseexponent = 3rd expression.

      2.  Convert to log form

      3.  Solve for the variable

      4.  It may be necessary to apply the change of base theorem to get a numerical/calculator result.

 

 

9.6  Applications

 

 

Ex 6:  How much stronger is one earthquake than another?   Find T for each, then divide:

      Answer:  the big one is ___ times greater than the little one. (in terms of shockwave)

Ex 8:  pH is measure of acidity (lower than 7, "acidic", greater than 7, alkaline, or "basic")

      Formula:  pH = -log[H+], where [H+] is the concentration of hydrogen ions in moles per liter.

 

 

Application:  solving interest equation for time (in particular, finding the doubling time or tripling time for an investment):  Ex 2, #62

 

Application:  Given a population equation (typically, exponential), find when the population will reach a certain number:  Ex 4, #67

 

 

 

 

Return to:  Merced College; Don Power               Updated 11/03/08 by Don Power