Session 12
Agenda:
• Questions from 6.1-6.3?
• 7.1 – Exponential Functions
• 7.2 – Logarithmic Functions
• Things to do before our next meeting.
Questions?
7.1 – Exponential Functions
• An exponential function is a function of the form
f ( x)  a x ,
a  0, a  1
• The number a is called the base of the exponential
function.
x
• Given the exponential function f ( x)  3 , evaluate the
following:
f (0)
f (1)
f (2)
f (1)
f (2)
Using the values you just found, sketch a graph of the
function.
f (0)  1
f (1)  3
1
3
1
f (2) 
9
f (1) 
f (2)  9
•
Domain:__________
•
Range:___________
•
What happens as x∞?
f(x) ______
•
What happens as x-∞?
f(x)______
•
Are there any asymptotes?
x
• In general, every basic exponential function f ( x)  a where
a>1 has the same properties and general shape.
• y-intercept of (0,1)
• No x-intercept
• Horizontal asymptote: y=0
• No vertical asymptote
• Domain: (-∞,∞)
• Range: (0,∞)
• Always increasing.
• As x∞, f(x) ∞
• As x-∞, f(x) 0
• The most commonly used exponential function is the
natural exponential function, which has base e.
f ( x)  ex
• All the properties from before still
hold for this function.
• The number e is an irrational number
whose value is between 2 and 3. The
graph of e x is shown in red between
the graphs of 2 x and 3x.
x
•
1
Consider the function f ( x)   . Evaluate the following and sketch a
2
graph.
f (0) 
f (1) 
f (1) 
f (2) 
f (2) 
•
Domain:__________
•
Range:___________
•
What happens as x∞?
f(x) ______
•
What happens as x-∞?
f(x)______
•
Are there any asymptotes?
• In general, every basic exponential function f ( x)  a where
0<a<1 has the same properties and general shape.
x
• y-intercept of (0,1)
• No x-intercept
• Horizontal asymptote: y=0
• No vertical asymptote
• Domain: (-∞,∞)
• Range: (0,∞)
• Always decreasing.
• As x∞, f(x) 0
• As x-∞, f(x) ∞.
• Consider the function f ( x)  2 x . Sketch a graph using a
x
transformation of the parent function 2 .
• This is the
same graph as x
x
1 1
1
f ( x)    since 2 x  x    .
2 2
2
• This is a decreasing exponential
function.
x
• The graph of f ( x)  e is also a reflection of e x across the
y-axis and is a decreasing exponential function. Its graph
x
x
is shown below in red between the graphs of 2 and 3 .
x
•
1
Consider the function f ( x)    . What is this graph
 3
equivalent to?
• Is this an increasing or decreasing exponential function?
• In general, a function of the form
f ( x)  a x ,
• Is decreasing if a>1.
• Is increasing if 0<a<1.
a  0, a  1
Sketch graphs of the following exponential functions using
transformations. Identify any asymptotes and whether the
function is increasing or decreasing.
x
f ( x)  2e  4
1
g ( x)  1   
 3
x 2
• Due to the fact that exponential functions are one-to-one,
the following is true:
If a x  a y , then x  y
• This fact can be used to solve exponential equations
where both sides can be expressed with a common base.
• Solve the following equations.
25 x  16
 1 
92 x   
 27 
x 1
1
25(53 x )   
5
x
7.2 – Logarithmic Functions
• Since exponential functions are one-to-one, they have
x
inverse functions. The inverse function of f ( x)  a (a  0, a  1)
is the logarithmic function with base a.
f ( x)  loga ( x),
defined by
a  0, a  1
y  loga ( x)  a y  x
• A logarithm is the EXPONENT to which the base must be
raised to get x.
• Evaluate the following:
log 2 16 
1
log3   
 81 
log100 1 
log9 3 
log5 0 
log 7 (49) 
• Important: You can ONLY take the logarithm of positive
numbers.
• Solve for x:
5x  log4 64  9
log6 x  4  7
•
Given f ( x)  log3 x, evaluate the following and sketch a graph.
1
1
log3   
log3   
9
 3
log3 (1) 
log3 (3) 
•
log3 (9) 
Domain:__________
•
Range:___________
•
What happens as x∞?
f(x) ______
•
What happens as x0+?
f(x)______
•
Are there any asymptotes?
• In general, every basic logarithmic function f ( x)  loga x
where a>1 has the same properties and general shape.
• x-intercept of (1,0)
• No y-intercept
• No Horizontal asymptote
• Vertical asymptote x=0
• Domain: (0,∞)
• Range: (-∞,∞)
• Always increasing.
• As x∞, f(x) ∞
• As x0+, f(x) -∞
• The logarithmic function with base 10 is called the
common logarithmic function and is usually just written
as f ( x)  log( x) .
• A special logarithmic function is the natural logarithmic
function which has base e. Instead of writing f ( x)  loge ( x) ,
we write this function as f ( x)  ln( x) .
• The graphs of both these functions have the same general
shape as any logarithmic function with base >1.
• Cancellation Properties:
log a (a x )  x
a loga x  x
ln(e x )  x
eln( x )  x
Evaluate the following, if possible.
ln(e)
log 4 (41000 )
ln(1)
log(1)
1
ln  2 
e 
 1 
log 

100


5log5 12
log 64 (e ln 4 )
Sketch graphs of the following functions using
transformations. Identify any asymptotes and the domain of
the function.
f ( x)   ln( x  2)
f ( x)  log2 ( x)  3
Find the domains of the following functions.
f ( x)  log(5x  9)
g ( x)  ln(8  2 x)  ln( x  9)
h( x ) 
3
log 2 x  2
Find the domain and range of the following function.

f ( x)  ln 4  x  1

Properties of Logarithms
log a ( xy )  log a x  log a y
x
log a    log a x  log a y
 y
log a ( x c )  c log a x
• Expand the following completely. Evaluate any known
logarithms.
 9 x2 
log 3 
 y 


  e( x  y ) 2 
ln   2 7  
 x y  


• Write the following as a single logarithm.
1
log( y )  3log(2 x)  log( x  y )
3
• Given that logb 2  n, logb 3  k , logb 7  w, write logb 504 in terms
of n, k, and w.
Change of Base Formula
• Any logarithm can be converted to a logarithm with
another base using the following formula.
log a x
logb x 
log a b
• Write log12 x using natural logarithms.
• If log4 x  2.8 , find log2 x .
Things to Do Before Next Meeting:
• Work on Sections 7.1-7.2 until you get all green
bars!
• Write down any questions you have.
• Continue working on mastering 6.1-6.3. After
you have all green bars on 6.1-6.3, retake the
Chapter 6 Test until you obtain at least 80%.