Transcript 11.3 & 11.6 Notes The natural number e and solving base e exponential equations.
Slide 1
11.3 & 11.6 Notes
The natural number e and solving
base e exponential equations
Slide 2
11.3 & 11.6 Notes
In this unit of study, you will learn several
methods for solving several types of
exponential equations.
In previous lessons, you learned how to solve
exponential equations using properties of
exponents and using logarithms.
In this lesson, you will learn how to solve base
e exponential equations.
Slide 3
11.3 & 11.6 Notes
x
1
Evaluate 1 for:
x
2 .7 0 5
a. x = 100
b. x = 1000
2 .7 1 7
c. x = 10000 2 .7 1 8
(round answers to the nearest thousandth)
Slide 4
11.3 & 11.6 Notes
x
1
e lim 1 2 .7 2
x
x
This is called the natural number e or Euler’s
number. It is named for the Swiss mathematician
Leonhard Euler for his work in the area of
logarithms.
Slide 5
11.3 & 11.6 Notes
Similar to π, e is an irrational number. That is, it
cannot be expressed as the ratio of integers. Its
value is a non-repeating, non-terminating decimal.
Similar to π, e is a naturally-occurring mathematical
phenomenon that cannot be completely explained.
We know that the ratio of every circle’s
circumference to its diameter is approximately 3.14.
Similarly, some exponential growths and decays
occur to a base of approximately 2.72.
Slide 6
11.3 & 11.6 Notes
Just as there are exponential equations with integers
and rational numbers as bases, there are exponential
equations with irrational numbers such as the
natural number e as bases.
How are base e exponential equations solved?
Using logarithms.
Slide 7
11.3 & 11.6 Notes
Since e is an irrational number, what base logarithm
will be used to solve base e exponential equations?
log e
Do you see a log base e button on your calculator?
Logarithms with the natural number e as their base
are called natural logarithms. Natural logarithm is
abbreviated ln.
ln log
e
Slide 8
11.3 & 11.6 Notes
e 5
x
ln e ln 5
x
x ln e ln 5
x 1 .6 0 9
ln log e
ln e log e e 1
Slide 9
11.3 & 11.6 Notes – Example 1
e 6
x
ln e ln 6
x
x ln e ln 6
x 1 .7 9 2
Slide 10
11.3 & 11.6 Notes – Practice 1
e 7
x
ln e ln 7
x
x ln e ln 7
x 1 .9 4 6
Slide 11
11.3 & 11.6 Notes – Example 2
1525 6 e
1525
e
4x
4x
6
1525
4x
ln
ln e
6
5 .5 3 8 4 x ln e
x 1 .3 8 5
Slide 12
11.3 & 11.6 Notes – Practice 2
1249 175 e
1249
e
3 x
3 x
175
1249
3 x
ln
ln e
175
1 .9 6 5 3 x ln e
x 0 .6 5 5
Slide 13
11.3 & 11.6 Notes – Example 3
14 6 e
2x
20
6e
2x
6
e
2x
1
ln e
2x
ln 1
2 x ln e 0
x0
Slide 14
11.3 & 11.6 Notes – Practice 3
3 x
8
3 x
12
3 x
6
3 x
ln 6
4 2e
2e
e
ln e
3 x ln e 1 .7 9 2
x 0 .5 9 7
Slide 15
11.3 & 11.6 Notes – Example 4
e
e
x
2x
7 e 10 0
x
2e 5 0
x
e 20
e 5 0
e 2
e 5
ln e ln 2
ln e ln 5
x ln e ln 2
x ln e ln 5
x
x
x
x 0.693,
x
x
x
1.609
Slide 16
11.3 & 11.6 Notes – Practice 4
e
e
x
2x
6e 8 0
x
2 e 4 0
x
e 20
e 40
e 2
e 4
ln e ln 2
ln e ln 4
x ln e ln 2
x ln e ln 4
x
x
x
x 0.693,
x
x
x
1.386
11.3 & 11.6 Notes
The natural number e and solving
base e exponential equations
Slide 2
11.3 & 11.6 Notes
In this unit of study, you will learn several
methods for solving several types of
exponential equations.
In previous lessons, you learned how to solve
exponential equations using properties of
exponents and using logarithms.
In this lesson, you will learn how to solve base
e exponential equations.
Slide 3
11.3 & 11.6 Notes
x
1
Evaluate 1 for:
x
2 .7 0 5
a. x = 100
b. x = 1000
2 .7 1 7
c. x = 10000 2 .7 1 8
(round answers to the nearest thousandth)
Slide 4
11.3 & 11.6 Notes
x
1
e lim 1 2 .7 2
x
x
This is called the natural number e or Euler’s
number. It is named for the Swiss mathematician
Leonhard Euler for his work in the area of
logarithms.
Slide 5
11.3 & 11.6 Notes
Similar to π, e is an irrational number. That is, it
cannot be expressed as the ratio of integers. Its
value is a non-repeating, non-terminating decimal.
Similar to π, e is a naturally-occurring mathematical
phenomenon that cannot be completely explained.
We know that the ratio of every circle’s
circumference to its diameter is approximately 3.14.
Similarly, some exponential growths and decays
occur to a base of approximately 2.72.
Slide 6
11.3 & 11.6 Notes
Just as there are exponential equations with integers
and rational numbers as bases, there are exponential
equations with irrational numbers such as the
natural number e as bases.
How are base e exponential equations solved?
Using logarithms.
Slide 7
11.3 & 11.6 Notes
Since e is an irrational number, what base logarithm
will be used to solve base e exponential equations?
log e
Do you see a log base e button on your calculator?
Logarithms with the natural number e as their base
are called natural logarithms. Natural logarithm is
abbreviated ln.
ln log
e
Slide 8
11.3 & 11.6 Notes
e 5
x
ln e ln 5
x
x ln e ln 5
x 1 .6 0 9
ln log e
ln e log e e 1
Slide 9
11.3 & 11.6 Notes – Example 1
e 6
x
ln e ln 6
x
x ln e ln 6
x 1 .7 9 2
Slide 10
11.3 & 11.6 Notes – Practice 1
e 7
x
ln e ln 7
x
x ln e ln 7
x 1 .9 4 6
Slide 11
11.3 & 11.6 Notes – Example 2
1525 6 e
1525
e
4x
4x
6
1525
4x
ln
ln e
6
5 .5 3 8 4 x ln e
x 1 .3 8 5
Slide 12
11.3 & 11.6 Notes – Practice 2
1249 175 e
1249
e
3 x
3 x
175
1249
3 x
ln
ln e
175
1 .9 6 5 3 x ln e
x 0 .6 5 5
Slide 13
11.3 & 11.6 Notes – Example 3
14 6 e
2x
20
6e
2x
6
e
2x
1
ln e
2x
ln 1
2 x ln e 0
x0
Slide 14
11.3 & 11.6 Notes – Practice 3
3 x
8
3 x
12
3 x
6
3 x
ln 6
4 2e
2e
e
ln e
3 x ln e 1 .7 9 2
x 0 .5 9 7
Slide 15
11.3 & 11.6 Notes – Example 4
e
e
x
2x
7 e 10 0
x
2e 5 0
x
e 20
e 5 0
e 2
e 5
ln e ln 2
ln e ln 5
x ln e ln 2
x ln e ln 5
x
x
x
x 0.693,
x
x
x
1.609
Slide 16
11.3 & 11.6 Notes – Practice 4
e
e
x
2x
6e 8 0
x
2 e 4 0
x
e 20
e 40
e 2
e 4
ln e ln 2
ln e ln 4
x ln e ln 2
x ln e ln 4
x
x
x
x 0.693,
x
x
x
1.386