Download Chapter 10 ISG

Interactive Study Guide for Students
Chapter 10: Exponential and Logarithmic Functions
Section 1: Exponential Functions
Exponential Functions
In an _________________ _________________, the base is a
constant and the exponent is a variable. In general, it is an
equation of the form y = abx, where a≠0, b>0 and b≠1. They have
the following characteristics:

The function is _________ and ___-to-____.

The domain is the set of all _________ numbers.

The x-axis is an _____________ of the graph.

The range is the set of all __________ numbers if a>0 and
all __________ numbers if a<0.

The graph contains the point ( , ). That is, the y-intercept
is a.

The graphs of y = abx and y = a(1/b)x are __________
across the ___-axis
Examples
1. Graph y = 2x. State the
domain and range.
Determine whether each
function represents exponential
growth or decay.
2. y = (1/5)x
3. y = 3(4)x
There are two types of exponential functions:
4. y = 7(1.2)x
____________ _______________ when the base is a number
greater than one.
Simplify:
____________ ____________ when the base is a number
between 0 and 1.
5. 2
5
2
 
6. 7
Exponential Equations and Inequalities
2
3
3
Solve:
The __________ of _____________ for Exponential Functions is
useful in solving the problems. It says: if b is a positive number
other than 1, then bx = by iff x = y.
7. 32n+1 = 81
Example: If 2x = 28 then x=8.
8. 42x = 8x-1
Property of ____________ for ____________ Functions: If b>1,
then bx>by iff x>y, and bx<by iff x<y Ex: If 5x<5y then x<y.
9. 43p-1>
1
256
Chapter 10: Exp. and Log. Functions
Section 2: Logarithms and Logarithmic Functions
Logarithms and Logarithmic Functions
To better understand logarithms and logarithmic functions, lets look
at the graph of y=2x and it’s inverse x=2y.
Examples
Write each equation in
exponential form.
1. log81=0
2. log2
1
=-4
16
Write each equation in
logarithmic form.
3. 103=1000
Since exponential functions are one-to-one, the inverse exists and are
also functions. Notice that the graph of these two functions are
reflections over the line y=x.
In general, the inverse of y=bx, is x=by and is called the
_______________ of x. It is usally written as y=logbx and is read y
equals log base b of x.
The function y=logbx, where b>0 and b≠1, is called the
___________________ ___________________. It has the following
characteristics:

The function is _________ and one-to-one.

The domain is the set of all _________ _____ numbers.

The y-axis is the _________ of the graph.

The range is the set of all __________ numbers

The graph contains the point (1,0), the _________ is 1.
1
4. 9 2 =3
Evaluate.
5. log264
6. log668
log  4 x 1
7. 3 3
8. log 4 n 
5
2
Solve Log. Equations and Inequalities
Use the definition of a logarithm to solve equations and inequalities.
If b>1, x>0 and logbx>y, then x>by
Solve:
If b>1, x>0 and logbx<y, then 0<x<by
9. log5x<2
Chapter 10: Exp. and Log. Functions
Section 2: Logarithms and Logarithmic Functions
Solve Log. Equations and Inequalities
Property of _________ for Log. Functions:
Examples


9. Solve log 5 p 2  2  log 5 p
If b is a positive number other than 1, then
log b x  log b y iff x  y .
Property of ____________ for Log. Functions:
If b>1, then log b x  log b y iff x>y and
10. Solve log 10 (3x  4)  log 10 ( x  6)
log b x  log b y iff x<y
*Remember than for log x, x cannot be negative*
Chapter 10: Exp. and Log. Functions
Section 3: Properties of Logarithms
Properties of Logarithms
Examples
1. If log 2 3  1.5850 ,
approximate the value of
log 2 48 .
____________Property of Logarithm:
For all positive numbers m, n, and b, b≠1,
log b mn  log b m  log b n
____________Property of Logartithm:
2. Use log 3 5  1.4650 and
For all positive numbers m, n, and b, b≠1,
log b
log 3 20  2.7268 to approx.
log 3 4 .
m
 log b m  log b n
n
____________Property of Logartithm:
For any real number p and positive numbers m and b, b≠1,
log b m p  p log b m
3. Given log 4 6  1.2925
approx. log 4 36 .
Solve:
Solve Logarithmic Equations
4.
Use the properties of logarithms to solve equations involving
logarithms.
3 log 5 x  log 5 4  log 5 16
5. log 4 x  log 4 ( x  6)  2
Chapter 10: Exponential and Logarithmic Functions
Section 4: Common Logarithms
Common Logarithms
Base ____ logarithms are called __________ logarithms. They can
be written without the subscript, and you can use the
__________________ to solve them.
Examples
Use the calculator to evaluate
each expression to four decimal
places.
1. log 3
2. log 0.2
3. Earthquakes are measured on
the Richter scale magnitude M by
log E  11.8  1.5M . In Feb.
2010, Chile recorded an
earthquake of 8.8. How much
energy was released?
4. Solve 3x=11
5. Solve 53y<8y-1
6. Express log 4 25 in terms of
Change of Base Formula
For all positive numbers a, b, and n, where a≠1 and b≠1,
common logarithms, then
approx. its value to four decimal
places.
log a n  --------------
Chapter 10: Exponential and Logarithmic Functions Section 5: Base e and Natural Log.
Base e and Natural Logarithms
1
n
For the expression 1(1  ) n (1) , as n increases, it approaches the
irrational number 2.71828… . This number is referred to as the
_________ ______, e . An exponential function with base e is
called a ___________ _______ ______________ __________.
Natural base exponential functions are used extensively in science
to model quantities that ______ and _______ continuously. Graph
y  ex :
Examples
Use a calc. to evaluate to four
decimal places:
1. e 2
2. e 1.3
3. ln 4
4. ln 0.05
Write an equivalent exp. or log.
function:
5. e x  5
6. ln x  0.6931
Evaluate:
7. e ln 7
8. ln e 4 x  3
The logarithm with base e is called the __________ ____________
sometimes denoted log e x but is more often abbreviated
_______. The ___________ ____________ ____________,
Solve:
y  ln x , is the inverse of the natural base exponential function
x
5e
y  e x . Graph y  ln x above.
Equations and Inequalities with e and ln
Equations and Inequalities involving base e are easier to solve
using _________ logarithms than using ___________ logarithms.
All of the properties of logarithms that you have learned apply to
natural logarithms as well.
When interest is compounded continuously, the amount A in an
account after t years is found using the formula
______________where P is the amount of principal and r is the
annual interest rate.
9.
7  2
10. ln 5x  4
11. ln( x  1)  2
Suppose you deposit $1000 in an
account paying 5%annual
interest.
12. What is the
balance after 10 years?
13. How long before the balance
is $1500?
Chapter 10: Exp. and Log. Functions
Exponential Decay
Section 6: Exponential Growth and Decay
Examples
When a quantity ___________ by a fixed percent over time, the
________ y of that quantity after t ______ is given by
y  a(1  r ) t where a is the _______ amount and r is the
_________ of decrease expressed as a __________. The percent
of decrease r is also referred to as the _________ ____
__________.
Another model for ____________ decay is y  ae  kt where k is a
__________. This is the model preferred by scientists, for example
in radioactive decay.
1. A caffeinated drink, containing
130 mg of caffeine, is eliminated
from the body at a rate of 11%
per hour. How long will it take
for ½ of it to be eliminated?
The half-life of a radioactive subs.
is the time it takes for half of the
atoms of the substance to
become disintegrated. The halflife of Carbon-14 is 5760 years.
2. What is the value of k for
Carbon-14?
3. A woolly mammoth contains
only 3% as much Carbon-14.
How long ago did the mammoth
die?
Exponential Growth
When a quantity ___________ by a fixed percent over time, the
________ y of that quantity after t ______ is given by
y  a(1  r ) t where a is the _______ amount and r is the
_________ of increase expressed as a __________. The percent of
increase r is also referred to as the _________ ____ __________.
Another model for ____________ growth is y  ae kt where k is a
__________. This is the model also preferred by scientists.
4. As of 2000, the pop. of China
was  1.26 billion, & India was 
1.01 billion. If I (t )  1.01e 0.015t
and C (t )  1.26e 0.009t , when will
India’s pop. be more than
China’s?