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Math AnalysisNotes
4.3 – Exponential Functions
4.4 – Logarithmic Functions
Date _______
An exponential function is a function in the form, f ( x)  a x , where a is a positive real number,
a > 0, a ≠ 1. The domain of f is the set of real numbers, (, ) .
The Laws for Exponents are true for real exponents:
 am  an  amn

a 
m n
 a mn

abn  a nbn `

1m  1

a  n 

a0  1
1
an
Example 1: Graph the exponential function, f ( x)  2 x .
x
y
Properties of exponential functions, f ( x)  a x , a > 1:
 Domain (, ) , Range (0, )
 No x-intercepts, y-intercept is (0, 1)
 x-axis is a horizontal asymptote as x → ─ ∞
 f ( x)  a x is an increasing function and is one – to – one
1
 Graph contains points (0, 1), (1, a), and (─1, )
a
 Graph is smooth, continuous, no corners or gaps
x
1
Example 2: Graph the exponential function, f ( x)    .
2
x
y
Properties of exponential functions, f ( x)  a x , 0 < a < 1:
 Domain (, ) , Range (0, )
 No x-intercepts, y-intercept is (0, 1)
 x-axis is a horizontal asymptote as x → ∞
 f ( x)  a x is a decreasing function and is one – to – one
1
 Graph contains points (0, 1), (1, a), and (─1, )
a
 Graph is smooth, continuous, no corners or gaps
n
 1
The number e is defined as the number that the expression 1   approaches as n →∞.
 n
n
 1
In calculus, the notation is expressed as lim 1    e .
n 
 n
Exponential equations are equations that involve terms of the form a x , a  0, a  1 . They are
solved using Laws of Exponents and the property, If a x  a y , then x  y .
Example 3: Solve the equation:
A.) 3x  2  27
B.) e x  e x  
2
3
1
e2
Since all one – to – one functions have inverse functions, the exponential function, f ( x)  a x ,
a > 0, a ≠ 1, has an inverse. The inverse of the exponential function is called the logarithmic
function.
The logarithmic function, y  log a x , is read “y is the logarithm of x to the base of a”, and is
defined by y  log a x if and only if x  a y .
The domain of the logarithmic function is (0, ∞). (The range of the exponential function!)
The range of the logarithmic function is (, ) . (The domain of the exponential function!)
Example 4: Change each exponential to logarithmic form.
A.) 43  a
B.) b3  27
C.) e x  1
Example 5: Rewrite each log in exponential form and solve for x.
A.) 4  log 3 x
B.) x  log 5 125
C.)  1  log x 6
To find the exact value of a log, we write the log in exponential form and then use the property,
If a m  a n then m  n.
Example 6: Find the exact value of each log.
A.) log 2 32
B.) log 5
1
125
To graph the logarithmic function, remember that it is the inverse of the exponential function,
and will reflect across the line y = x.
Example 7: Graph the function, y  log 2 x.
Properties of the graph of log functions, y  log a x.
 Domain (0, ) , Range (, )
 x-intercept is (1, 0), no y-intercept
 y-axis is a vertical asymptote

y  log a x. is an increasing function if a > 1 and decreases if 0 < a < 1
1
 Graph contains points (1, 0), (a, 1), and ( ,─1)
a
 Graph is smooth, continuous, no corners or gaps
Example 8: Find the domain of each log function:
A.)
f ( x)  log 2 ( x  4)
B.)
 x  2
g ( x)  log 5 

 x2
Special logarithm functions:

The common logarithm function is the log function with base 10.
When the base is not indicated, it is understood to be 10.
y  log x , if and only if 10 y  x

The natural logarithm function is the log function with base e. It is written with a
special symbol ln, instead of log.
y  log e x  ln x , if and only if e y  x
Example 9: Use your calculator to determine the value of each log to three decimal places:
1
A.) log 8
B.) log
C.) ln 5
5
Example 10: Use transformations to graph the function, f ( x)  2 log( x  1).
Determine the domain, range, and vertical asymptote of the graph.
Solving logarithmic equations:
 Isolate the log
 Convert to exponential form
 Solve the exponential equation
 Check for extraneous solutions
Example 11: Solve each equation:
A.) log 4 (2x  1)  3
B.) log x 81  2
Using logs to solve exponential equations:
 Change to a log expression
 Solve for the variable
 Use calculator to approximate the solution
Example 12: Solve the equation,
e3 x  6
Assignment: pp. 255 – 260: # 29 – 36, 37, 49, 53, 57, 62
pp. 269 – 271; # 21, 31, 33, 35, 36, 42, 45, 47, 53,
67-74, 75, 89, 91, 93, 103, 107
Math Analysis Notes
4.5 – Properties of Logs
4.6 – Log and Exponential equations
Date _______
Properties of Logs:
 log b 1  0
 log b b  1

log b b m  m


b logb M  M
Log of a product: log b (MN )  log b M  log b N
M
 log b M  log b N
Log of a quotient: log b
N
Log of a power: log b M n  n log b M



M = N if and only if log b M  log b N
Example 1: Writing a log expression as a sum, difference, or power of logs –
AKA Expanding a log expression
A.) log a xy2 
C.) log a
x
( x  1)

B.) log a x x  3
D.) ln

x x 1
( x  2)3
Example 2: Writing log expressions as single logs – AKA condensing a log expression
A.) log a 4  3 log a x
C.) 2 ln x  ln 10  ln( x 2  1)  ln 3
B.)
1
log( x  1)  3 log x
2
Warning! Be careful not to express the log of a sum as the sum of logs!
For example, log a ( x  2)  log a x  log a 2 .
Example 3: Use the properties of logs to find the value of log 3 8 .
Round to three decimal places.
Change of base Formula:
If a  1, b  1 , and M are positive real numbers, then
log b M
log a M 
log b a
Example 4: Use the change of base formula to find the value of each log. Approximate answer
to three decimal places
A.) log 6 42
B.) log
3
 7
Solving a log equation:
 Determine the domain of the individual logs - combine to find the domain of the
expression
 Use properties to condense to a single log
 Convert to exponential form
 Solve the resulting equation
 Check for extraneous solutions
Example 5: Solve the log equation.
A.) log 4 8  2 log 4 x
B.) ln( x  1)2  2
C.) log( x  1)  log( x  2)  1
D.) log2 (x - 2) - log4 x =1
(Hint: Change log 4 x to base 2).
Solving exponential equations:
If bases are equal, use a x  a y  x  y
If bases are not equal, then
 Isolate the exponential
 Convert to logs (common or natural usually best
 Solve for the variable
 Use calculator to approximate solution
Example 6: Solve each equation. Round answers to three decimal places.
A.) 4 x  10
B.) 4 2 x 1  27  0
C.) 4 x 1  32 x 1
D.)
e x  e x
3
2
Assignment: pp. 280 – 282; # 1 – 6, 9, 13, 17, 23, 24, 39, 41, 51, 53, 61, 65, 85
pp. 286 – 287; # 1, 3, 9, 11, 13, 21, 29, 34, 35, 37, 42
Math Analysis Notes – 4.7 and 4.8
Applications of Logs and Exponentials
Date _______
Simple interest: Interest calculated once at end of time period
P = principal (dollars)
t = time (years)
r = interest rate (per cent per year converted to decimal)
I = amount of interest
Simple Interest formula: I  Prt
The amount A that must be repaid: A  P  I or A  P(1  rt )
Payment periods:
Annually = _____ times per year
Quarterly = _____ times per year
Daily = _____ times per year
Semi-annually = _____ times per year
Monthly = _____ times per year
Compound Interest: When interest due at the end of the period is added to the principal so that
the interest computed at the end of the next period will include the new principal amount.
Compound interest: P = principal (dollars)
t = time (years)
r = interest rate (per cent per year converted to decimal)
n = number of compoundings per year

Compound interest formula: A  P 1 

rt
Continuous compounding: A  Pe
r

n
nt
Example 1: Rudolph invested $1000 in an account that pays 6.5 % interest. How much money
will he have in the account after 5 years if it compounded semi-annually?
Find the amount if interest is compounded continuously.
Example 2: Wendy has $2000 to invest. Her account pays interest at a rate of 5%, compounded
daily. How long will it take for her investment to double?
Exponential Growth or Decay: When values changes exponentially (rather than linearly)
A(t)  A0 e kt where A0 = initial amount
A(t) = amount after t years
k = constant ( k  0)
A positive value for k indicates growth, a negative value indicates decay
Example 3: A number of bacteria N present in a culture at time t (hours) grows according to the
formula N(t)  1000e0.01t .
a) What is the population after 4 hours?
b) When will the number of bacteria reach 1700?
c) When will the number of bacteria double?
Half – life: The amount of time required for half of an amount of a substance to decay
Example 4: The half-life of radioactive carbon is 5600 years. If 100 grams of carbon is present
in an object now, how much will be present in 100 years?
How much will be present in 1000 year?
Assignment: pp. 294 – 295: # 3, 7, 9, 13, 27, 31
pp. 304 – 305: #1, 3, 5, 7, 11, 13
And REVIEW FOR THE QUIZ!!!!!!!!!!!!!!