Transcript Document

Tuesday, 12 April 2016
The Exponential and Logarithmic
Functions.
Objectives:
Recognise the graphs of exponential functions.
Know how to plot and sketch the graphs of exponential
functions.
Know how to solve equations involving exponents and
logarithms.
Objectives
Solve equations involving exponents
and logs.
Solve more complex equations involving
exponents and logs.
Solve problems in context using
exponents and logs.
y = ln x
y=
ex
y = e-x
y = -e-x
y = 3ex
y = log x
y = 1+e2x
y = e2x
y = 2ln x
Problem 1
ex = 10
ln ex = ln 10
x ln e = ln 10
x = ln 10
Problem 2
ln x = 5
elnx = e5
x = e5
Problem 3
e2x + 3 = 4
ln e2x + 3 = ln 4
(2x+3) ln e = ln 4
2x+3 = ln 4
x = ½((ln 4) – 3)
Problem 4
2 ln x + 1 = 5
2 ln x = 4
ln x = 2
x = e2
Problem 5
The number of elephants in the herd can be represented
by the equation : N  150  80e

t
40
(t is the time in years after 2003)
Calculate the number of elephants in the herd in (a) 2003
(b) 2007.
(c) Find the year when the population will rise above 100.
(d) Find the long term population predicted by the model.
Sketch a graph of N against t.
Calculate the number of elephants in the herd in (a) 2003
When t = 0, N = 70
Calculate the number of elephants in the herd in (b) 2007
When t = 4, N = 78 (to the nearest elephant!)
t = 18.8 so the population will be over 100 in 2022.
As t tends to infinity, the population tends to 150
Graph
Example 1
• Solve:
e2x- 6ex = 7
Set y =
x
e
So, e2x = y2
The equation becomes ...
y2 - 6y = 7
y2 - 6y - 7 = 0
(y -7)(y + 1) = 0
Either: (y-7)=0 y=7
or:
(y+1)=0 y=-1
ex = 7
ex = -1
x = ln 7
x = ln -1
Example 2
• Solve:
5 - ex = 4e-x
Set y =
ex
So, e-x = 1/y
The equation becomes ...
5 - y = 4/y
5y - y2 = 4
y2 - 5y + 4 = 0
(y -4)(y - 1) = 0
Either: (y-4)=0 y=4
or:
(y-1)=0 y=1
ex = 4
ex = 1
x = ln 4
x = ln 1
Summary
When you have
single exponentials
e2x = e6
Log both sides
ln e2x = ln e6
2x ln e = 6 ln e
When you have logs
2 ln x = 8
ln x2 = 8
e ln x’ = e8
x2 = e8
When you have different
exponentials
e2x +4ex +3 = 0
y2 +4y +3 = 0
x + 1) = 0
(y + 3)(y
(ex + 3)(e + 1) = 0
Find the exponent of
both sides
You may need to
manipulate first
Use substitution
y = ex
Follow Up Work:
1. Exercise 3B – alternate letters from
each question.
2. Exercise 3C next lesson.
3. Chapter Assessment
x
e
y = ln x
y=
y = log x
y = e-x
y=1 +
2x
e
y=
2x
e
-x
y=-e
y = 3ex
y = 2ln x
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