MA 1128: Lecture 20

1
MA 1128 Lecture 20 4/21/11

• Exponential Functions
• Logarithmic Functions

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Exponential Functions

• There are a few special kinds of functions that
  are commonly used.
• Were going to talk about exponential and
  logarithmic functions in this class,
• and there are also trigonometric functions (like
  f(x) sin x) that we wont cover.
• Exponential functions have their variable in the
  exponent with a constant base.
• For example, f(x) 2x and f(x) 10x are
  exponential functions.
• Lets look at the exponential function f(x)
  2x.
• When we evaluate this function at whole number
  values, we get 2 multiplied by itself x
  times.
• For example, f(1) 21 2, f(2) 22 2?2
  4, and f(3) 23 2?2?2 8.

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Example (cont.)

• For other values of x, mathematicians have
  invented meanings for 2x.
• For example, we have already used negative
  exponents to mean one over,
• and f(-2) 2-2 1/22 1/4.
• We also have let anything to the zero power be
  one,
• so f(0) 20 1.

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4
Practice Problems

• For the same function f(x) 2x, do the
  following.
• Compute f(-3) and f(-1).
• We now have found function values for x -3,
  -2, -1, 0, 1, 2, and 3. Plot these points
  carefully and draw a nice smooth curve through
  them. In the quiz, youll be choosing between
  graphs that are subtly different, so do this
  carefully.
• Answers
• 1) 1/8 and 1/2.
• 2) There is a graph of this function on Slide 6.

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5
More on f(x) 2x.

• Another mathematical invention that weve been
  using with exponents is writing radicals with
  fractional exponents.
• For example, f(1/2) 21/2 1.414 (the square
  root of 2),
• and f(3/4) 23/4 1.681 (the fourth root of 2
  cubed).
• Of course, your calculator may know this last one
  as 20.75.
• In any case, if you plot these two points, you
  will see that they lie on the smooth curve you
  drew in problem 2 on the previous slide.
• This is evidence that our interpretation of
  negative and fractional exponents is a good one.
• Remember that your calculator knows the values
  for all of the exponential functions.
• The graph of f(x) 2x is shown on the next
  screen.

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Graph of f(x) 2x

• This graph is pretty typical of exponential
  functions.
• They always go through (0,1).
• This one approaches the x-axis towards the left
  and goes to infinity towards the right. Some may
  be backward from this and go to infinity on the
  left.

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7
Logarithmic Functions

• When we have solved equations, we have relied on
  one basic trick very heavily.
• We got rid of things by doing the opposite.
• For example, to get rid of the plus 3 in 2x
  3 7,
• we would subtract 3 from both sides.
• To get rid of the times 2 in 2x 4,
• we would divide by 2 on both sides.
• To get rid of the square in (x 2)2 9,
• we would take the square root of both sides.
• The logarithmic functions are precisely those
  functions that do the opposite of the
  exponential functions.

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Cont.

• For example, the log base 2 function g(x) log2
  x does exactly the opposite of the exponential
  function f(x) 2x.
• In terms of an equation, suppose we have 2x 7,
  and we want to solve for x.
• We have 2x on the left side, and we want to
  get rid of it.
• We should do the opposite, that is, take the
  log base 2 of both sides.
• log2 (2x) log2(7)
• x log2(7)
• Now, if we had a log2-button on our calculators,
  wed be done.
• Well say were done anyway, and call this the
  exact solution.
• Well talk about using our calculators later.

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Practice Problems

• Find the exact solution to the following
  equations.
• 2x 3.
• 5x 2. (use log5)
• 10x 3.
• Answers
• 1) x log2(3)
• 2) x log5(2)
• 3) x log10(3)

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Log Base 10, a.k.a. Common Logs

• In problem 5, you should have gotten x
  log10(3).
• You should have a log10-button on your
  calculator.
• It turns out that in any application of logs, you
  can get by with any one particular log function.
• For a long time, and in many situations, it was
  log10 that was used.
• Big tables with values for the log10 function
  were very common.
• Im guessing that thats why these logs are
  called common logs,
• and instead of writing log10(3), we usually just
  write log(3).
• Here, we drop the little 10,
• kind of like how we dont write the little 2 in a
  square root.

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Cont.

• The log10-button on your calculator probably has
  LOG or log on it.
• log10(3) log(3) .477121255 On your
  calculator 3, log.
• You can check this in the equation 10x 3.
• 10.477121255 3.000000002 on my calculator.
• The .000000002 is round-off error.
• We can use the log10-button on our calculator to
  compute any log.
• It turns out that log2(x) log(x)/log(2).
• We had log2(7) before.
• We can compute this with log(7)/log(2)
  2.807354922 7, log, ?, 2, log, .
• This was the solution to the equation 2x 7, so
  we can check this.
• 22.807354922 7. (Try this on your calculator!)

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Example

• There is a similar formula for any log, so its
  pretty easy to solve equations involving
  exponential functions.
• Suppose we want to solve the equation 5x 15.
• Take log5 of both sides
• log5(5x) log5(15)
• x log5(15) log(15)/log(5) 1.682606194.
• Rounded to 4 decimal places, this is 1.6826.

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Practice Problems

• Solve each equation, and write your answer in
  decimal form rounded correctly to 4 decimal
  places. 10x 100.
• 10x 20.
• 5x 35.
• 2x 20.73
• 4x 16.
• Answers
• 1) x 1.3010
• 2) x 2.2091
• 3) x 4.3736
• 4) x 2

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More about logs

• The graph of a log function looks just like the
  graph of the corresponding exponential function,
  but its turned over on its side.
• The top green graph is 3x, and the bottom red
  one is log3(x).

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Cont.

• If you read about the properties of logs in the
  optional textbook, or any book that talks about
  logs, youll find some basic rules about logs.
  Roughly, these say that multiplication, division,
  and exponentiation inside a log are equivalent to
  addition, subtraction, and multiplication
  outside.
• I wont say more about these, but you should be
  aware that these properties of logs exist, and if
  you think about them, you might see that they
  correspond to properties of exponents, and by
  bringing the level of computation down a level,
  they can make complicated arithmetic computations
  easier.

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