Algebra 2 Chapter 4 Note-Taking Guide
Exponential and Logarithmic Functions
4.1
Name ______________________
Per ____ Date _______________
Graph:
The graph from above is of the function: y  2x
The domain of the graph is
The range of the graph is
4.1
A graph with a variable as an exponent is known as an exponential function.
 The parent exponential function is f ( x)  b x , where the base ___ is a constant and the
exponent ___ is the independent variable.
 Note: the base must be positive (b > 0) and it cannot be one.
 The x-axis if the asymptote for the parent exponential function. An asymptote is a line that a
graphed function approaches (but never touches) as x gets very small or very large.
Exponential functions come in the form y  ab x
 If a  0 and b  1 , the function represents exponential growth (as x increases, y increases
exponentially)
 If a  0 and 0  b  1 , the function represents exponential decay (as x increases, y
decreases exponentially)
 All exponential functions will have a y-intercept of (0, a)
4.1
Tell whether the function shows growth or decay. Graph.
f(x) = 1.5x
g(x) = 30(0.8)x
4.1
Where will we see this in real-life?
You can model growth or decay by a constant percent increase or decrease with the following:
t
A(t )  a 1  r 





4.1
A(t) is the amount at any point of time (final amount)
a is the initial amount
r is the rate of increase/decrease (written as a decimal)
t is the number of time periods
MEMORIZE!
Tony purchased a rare 1959 Gibson Les Paul guitar in 2000 for $12,000. Experts estimate that its
value will increase by 14% per year. Use a graph to find when the value of the guitar will be $60,000.
 Write the equation

Graph on your calculator

Estimate the solution
4.1
In 1981, the Australian humpback whale population was 350 and has increased at a rate of about
14% each year since then. Write a function to model population growth. Use a graph to predict when
the population will reach 20,000.
4.1
The value of a truck bought new for $28,000 decreases 9.5% each year. Write an exponential
function, and graph the function. Use the graph to predict when the value will fall to $5000.
4.1
A motor scooter purchased for $1000 depreciates at an annual rate of 15%. Write an exponential
function, and graph to predict when the value will fall below $100.
4.2
We have seen the word inverse used in various ways:
 The additive inverse of 3 is -3

The multiplicative inverse of 5 is
1
5
You can also find and apply inverses to relations and functions. To graph an inverse relation, you
can reflect each point across the line y  x . This is the same as switching the x and y-values for each
ordered pair of the relation.
4.2
Graph the relation, identify the domain and range.
x 0 1 2 4 8
y 2 4 5 6 7
Find the inverse relation. Identify the domain and range.
Graph the relation
x
y
4.2
Functions that “undo” each other are inverse functions.
 Inverse functions have the notation: f ( x) and f 1 ( x)
 Inverse functions are reflections of each other over
the line y  x
Example: Graph y  2x and graph its inverse
**You can draw the inverse of any function on your
calculator by 2nd program, DrawInv, Vars, Y-vars, enter,
enter
4.2
Guidelines for finding the inverse:
1. Verify that your function is 1-to-1
 Vertical line test
 Horizontal line test
2. Substitute y in for f(x)
3. Switch your x and y in the equation
4. Solve for y
5. Substitute f-1(x) for y
4.2
Find the inverse function for each:
1.) f ( x)  2 x
2.) f ( x) 
x
5
4
3.) f ( x)  x 
2
3
4.2
Graph f ( x)  3x  6 . Write and graph the inverse.
4.3
What is a logarithm? A logarithm is an inverse operation that undoes raising a base to an exponent.
A logarithm is the exponent to which a specific base is raised to obtain a given value. It is the inverse
of the exponential function.
bx  a
4.3
log b a  x
Write each exponential as a log:
1.) 26  64
4.3
becomes
2.) 41  4
4.) 32 
3.) 50  1
More practice:
5.) 92  81
6.) 33  27
4.3
Write each logarithm as an exponential:
1.) log10 100  2
2.) log5 5  1
4.3
More practice
4.) log10 10  1
5.) log12 144  2
7.) 23 
1
8
3.) log12 1  0
6.) log 1 8  3
2
1
9
4.3
The common log is a logarithm with base 10 ( log10 ) and can be written simply as log .
Find the value of each by rewriting as an exponential function:
1.) log100
4.3
2.) log1000
Evaluate:
1.) log7 49
4.3
4.) log
3.) log1
2.) log 4
1
4
Because logarithms are the inverses of exponents, we can
graph logarithms by graphing the exponential function and then
using the inverse properties.
Graph f ( x)  3x
Graph f ( x)  log3 x
4.4
Properties of Logarithms:



4.4
logb mn  logb m  logb n
m
logb  logb m  logb n
n
logb mk  k logb m
Write each expression as a single logarithm:
1.) log 4 2  log 4 32
2
3.) log3 81
2.) log5 625  log5 25
1
4.) log 5  
5
3
1
100
4.4
Use the properties of logs to rewrite in terms of ln 2 and ln 3
1.) ln 6
2.) ln
4.4
Expand using the properties of logarithms:
1.) log10 5x3 y
4.4
Inverse properties of logarithms:


4.4
4.4
2
27
logb b x  x
blogb x  x
Simplify:
1.) log8 83 x1
2.) log5 125
3.) 2log2 27
4.) log3 311
5.) log3 81
6.) 5log5 10
_____ Which statement is not true?
(a) log140  log 35  log 4
log140
 log 4
log 35
(c) log 35  log 4  log140
140
 log 4
(d) log
35
(b)
4.4
Change of base formula:
log a x 
4.4
4.5
Calculate:
1.) log3 7
log b x
log b a
but since we have the common log we can use
log a x 
log x
log a
2.) log5 123
An exponential equation is an equation containing one or more expressions that have a variable as an
exponent. To solve exponential equations:
 Try rewriting them so the bases are all the same
 Using the inverse operation of logarithm to undo the exponential.
4.5
Solve for x:
1.) 8x  2 x 6
4.)
7  x  21
2.)
5.)
5x 2  200
3.)
32 x  27
23 x  15
4.5
Suppose a bacteria culture doubles in size every hour. How many hours will it take for the number of
bacteria to exceed 1,000,000?
4.5
1.) log3 ( x  5)  2
2.) log 45 x  log 3  1
3.) log6 (2 x  1)  1
4.) log 4 100  log 4 ( x  1)  1
5.) log5 x 4  8
6.) log12 x  log12 ( x  1)  1
4.6


In the past, you may have seen the compound interest formula: A  P 1 
r

n
nt
Where A is the total amount of money, P is the principal (starting value of money), r is the annual
interest rate, n is the number of times the interest is compounded per year, and t is the time in years.
Well, when n gets very large, interest is continually compounded. In fact if we were to graph
n
 1
f (n)  11   , we would see that the graph approaches the number 2.7182818284590 . . . .
 n
This number is called e. Like  , e is an irrational number.
4.6
Exponential functions with base e have the same properties as
other exponential functions.
Graph f ( x)  e x
4.6
The inverse of f ( x)  e x is the natural log ( ln x )
Simplify
1.) ln e 2 x
4.) ln e0.15 x
2.) eln( x 1)
3.) e5ln x
5.) e3ln( x 1)
4.6
The exponential model we use when compounding interest continuously is:
4.6
What is the total amount for an investment of $1000 invested at 5% for 10 years compounded
continuously?
4.6
What is the total amount for an investment of $100 invested at 3.5% for 8 years and compounded
continuously?
A  Pert
4.6
Half-life of a substance is the time it takes for half of the substance to breakdown or convert to
another substance during the process of decay. Natural decay is modeled by the function below:
4.6
A paleontologist uncovers a fossil of a saber-toothed cat in California. He analyzes the fossil and
concludes that the specimen contains 15% of its original carbon-14. Carbon-14 has a half-life of 5730
years. Use carbon-14 dating to determine the age of the fossil.
4.7
Transformation rules:

4.7
f ( x)  k
f ( x  h)

af ( x)
1 
f  x
b 

 f ( x)
f ( x)
Make a table of values, and graph the function
f ( x)  2 x  4 . Describe the asymptote. Tell how the
graph is transformed from the graph f ( x)  2x .
4.7
Make a table of values, and graph the function
f ( x)  2x 2 . Describe the asymptote. Tell how the graph
is transformed from the graph f ( x)  2x .
4.7
Graph the function f ( x)   ln( x  4) . Describe the
asymptote. Tell how the graph is transformed from the
graph f ( x)  ln x .
4.7
Graph the function f ( x)  3log x  5 . Describe the
asymptote. Tell how the graph is transformed from the
graph f ( x)  log x .
4.8
We have worked on linear regressions, quadratic regressions, and cubic regressions. Now we will
work on exponential and logarithmic regressions:
The table gives the approximate values of diamonds of the same quality. Find an exponential model
for the data. Use the model to estimate the weight of a diamond worth $2325.
4.8
Find an exponential model for the data. Use the model to predict
when the tuition at UT Austin will be $6000.
4.8
Use exponential regression to find a function that models this data. When will the number of bacteria
reach 2000?
4.8
Use logarithmic regression to find a function that models this data. When will the speed reach 8.0
m/s?