EXPONENTIAL FUNCTIONS
A function of the form y=abx,
where a=0, b>0 and b=1.
 Characteristics
1. continuous and one-to-one
2. domain is the set of all real numbers
3. Range is either all real positive numbers or all real negative numbers
depending on whether a is , or > 0
4. x-axis is a horizontal asymptote
5.y-intercept is at a
6. y=abx and y=a(1/b)x are reflections across the y-axis
EXAMPLE 1
Sketch the graph of y=2x. State the domain and range.
EXAMPLE 2
1
2
Sketch y=( )x. State the domain and range.
EXPONENTIAL GROWTH & DECAY
Exponential Growth:
Exponential function with base greater than one.
 y=2(3x)
Exponential Decay:
Exponential function with base between 0 and 1
 y=4(1/3)x
EXAMPLE 3-6
Determine if each function is exponential growth or decay
y=(1/5)x
y=7(1.2)x
y=2(5)x
y=10(4/3)x
STEPS TO WRITE AN
EXPONENTIAL FUNCTION
1. Use the y-intercept to find a
2. Chose a second point on the graph to substitute into the
equation for x and y. Solve for b.
3. Write your equation in terms of y=abx (plug in a and b)
EXAMPLE 5
Write an exponential function using the points (0, 3) and (-1, 6)
EXAMPLE 6
Write an exponential function using the points (0, -18) and (-2, -2)
EXAMPLE 11
In 2000, the population of Phoenix was 1,321,045 and it increased
to 1,331,391 in 2004.
 A. Write an exponential function of the form y=abx that could be
used to model the population y of Phoenix. Write the function in
terms of x, the number of years since 2000.
 B. Suppose the population of Phoenix continues to increase at the
same rate. Estimate the population in 2015.
EXPONENTIAL EQUATIONS
Exponential equation:
An equation in which the variables are exponents
Property of Equality
If the base is a number other than 1 and the base is the same , then
the two exponents equal each other.
 2x = 28 then x=8
STEPS TO SOLVE EXPONENTIAL
EQUATIONS/INEQUALITIES
1. Rewrite the equation so all terms have like bases (you may need
to use negative exponents)
2. Set the exponents equal to each other
3. Solve
4. Plug x back in to the original equation to make sure the answer
works
Solve
32n+1 = 81
EXAMPLE 3
Solve
35x = 92x-1
EXAMPLE 4
Solve
42x = 8x-1
EXAMPLE 7
Solve
4
3 p 1
1

256
EXAMPLE 11
Solve
5
2 x 3
 125
EXAMPLE 13
Solve
4
4a 6
16
a
Logarithms with base b
log b x  y
Say: “Log of x base b is y”
LOGARITHMIC TO EXPONENTIAL FORM
1. log 8 1  0
1
3. log 3
 3
27
2. log 4 16  2
EXPONENTIAL TO LOGARITHMIC FORM
4.a  1000
3
1
2
6 .9  3
5.4  64
3
EVALUATE LOGARITHMIC EXPRESSIONS
7. log 2 64
8. log 3 81
CHARACTERISTICS OF
LOGARITHMIC FUNCTIONS
1. Inverse of the exponential function y=bx
2.Continous and one-to-one
3. Domain is all positive real numbers and range is ARN
4. y-axis is an asymptote
5. Contains (1,0), so x-intercept is 1
HELPFUL HINT
Since exponential and logarithmic functions are
inverses if the bases are the same they “undo”
each other…
log 6 6  8
8
log3 ( 4 x 1)
3
 4x 1
LOGARITHMIC EQUATIONS
Property of Equality
 If b is a positive number other than 1, then log b x  log b y
if and only if x = y.
log 7 x  log 7 3
x3
EXAMPLE 9
Solve
5
log 4 n 
2
EXAMPLE 10
Solve
log 4 x 2  log 4 (4 x  3)
EXAMPLE 11
Solve
log 5 ( p  2)  log 5 p
2
LOGARITHMIC TO EXPONENTIAL INEQUALITY
If b > 1, x > 0 and
logbx > y then x > by
If b > 1, x > 0 and logbx < y
then 0< x < by
log 2 x  3
log 3 x  5
x2
0 x3
3
5
EXAMPLE 12
Solve
log 5 x  2
EXAMPLE 13
Solve
log 4 x  3
PROPERTY OF INEQUALITY FOR LOGARITHMIC
FUNCTIONS
If b>1, then log b
and
x  log b y
if and only if x>y
log b x  log b y if and only if x<y
EXAMPLE 14
log 10 (3x  4)  log 10 ( x  6)
EXAMPLE 15
log 7 (2 x  8)  log 7 ( x  5)
PRODUCT PROPERTY
The logarithm of a product is the sum of the
logarithm of its factors
log b (m)(n)  log b m  log b n
QUOTIENT PROPERTY
The logarithm of a quotient is the difference of the
logarithms of the numerator and denominator.
m
log b  log b m  log b n
n
POWER PROPERTY
The logarithm of a power is the product of the
logarithm and the exponent
log b m  p  log b m
p
EXAMPLE 1
3 log 5 x  log 5 4  log 5 16
EXAMPLE 2
log 4 x  log 4 ( x  6)  2
EXAMPLE 3
2 log 7 x  log 7 27  log 7 3
EXAMPLE 4
4 log 2 x  log 2 5  log 2 125
EXAMPLE 5
log 3 42  log 3 n  log 3 7
EXAMPLE 6
2 log 5 x  log 5 9
COMMON LOGARITHMS
Logarithms with base 10 are common logs
 You do not need to write the 10 it is understood
log 100
 Button on calculator for common logs
LOG
EXAMPLES: USE CALCULATOR TO EVALUATE
EACH LOG TO FOUR DECIMAL PLACES
1. log 3
2. log 0.2
3. log 5
4. log 0.5
SOLVE LOGARITHMIC EQUATIONS
Example 5:
The amount of energy E, in ergs, that an earthquake
releases is related to is Richter scale magnitude M by
the equation logE = 11.8 + 1.5M. The Chilean
earthquake of 1960 measured 8.5 on the Richter
scale. How much energy was released?
Example 6:
Find the energy released by the 2004 Sumatran earthquake, which
measured 9.0 on the Richter scale and led to the tsunami.
HELPFUL HINT
If both sides of the equation cannot be easily written as
powers of the same base you can solve by taking the
log of each side!
EXAMPLE
3x=11
4x=15
SOLVING INEQUALITIES
Example 7
53y<8y-1
EXAMPLE 8
32x>6x+1
EXAMPLE 9
4y<52y+1
CHANGE OF BASE FORMULA
log10 12
log 5 12 
log10 5
EXAMPLE
Express in terms of common logs, and then approximate its value to
four decimal places.
log425
log318
log7 5
NATURAL BASE EXPONENTIAL
FUNCTION
An exponential function with base e
 e is the irrational number 2.71828…
*These are used extensively in science to model quantities that
grow and decay continuously
Calculator button
ex
EVALUATE TO FOUR DECIMAL PLACES
1. e2
2. e-1.3
3. e1/2
THE LOG WITH BASE E IS A
NATURAL LOG
Written as : ln
y=ln x is the inverse of y = ex
All properties for logs apply the same way to natural logs
Calculator button
lnx
EXAMPLES
Use calculator to evaluate to four decimal places
4. ln4
5. ln0.05
6. ln7
EXAMPLE
Write an equivalent exponential or log equation to the given equation.
7. ex=5
8. lnx≈0.6931
REMEMBER…..
All log properties apply to natural logs
Do the same thing for ln problems that you do for log problems
Let’s solve!!!!!!!!!
EXAMPLE 9
Solve e4x=120 and round to four decimal places
EXAMPLE 10
EXAMPLE 11
ex-2 + 4<21
ln6x > 4
EXAMPLE 12
EXAMPLE 13
ln5x+ln3x>9
2e3x+5=2
EQUATIONS THAT DEAL WITH E
Continuously Compounded Interest
A=Pert
A= amount in account after t years
t= # of years
r= annual interest rate
P= amount of principal invested
EXAMPLES
Suppose you deposit $1000 in an account paying 2.5% annual
interest, compounded continuously.
 Find the balance after 10 years
 Find how long it will take for the balance to reach at least $1500
Suppose you deposit $5000 in an account paying 3% annual interst,
compounded continuously.
 Find what the balance would be after 5 years
 Find how long it will take for the balance to reach at least $7000
EXPONENTIAL DECAY
y=a(1-r)t
a=initial amount
r=% of decrease expressed as a decimal, this is also called rate of decay
t=time
EXPONENTIAL DECAY
y=ae-kt
a=initial amount
k=constant
t=time
EXAMPLE 3
A cup of coffee contains 130 milligrams of caffeine. If caffeine is
eliminated from the body at a rate of 11% per hour, how long will
it take for half of this caffeine to be eliminated?
EXAMPLE 4
The half-life of Sodium-22 is 2.6 years.
 What is the value of k and the equation of decay for Sodium-22?
 A geologist examining a meteorite estimates that it contains only
about 10% as much Sodium-22 as it would have contained when it
reached Earth’s surface. How long ago did the meteorite reach
Earth?
EXPONENTIAL GROWTH
y=a(1+r)t
a= initial amount
r=% of increase/growth expressed as a decimal
t=time
y=aekt
a=initial amount
k=constant
t=time
EXAMPLE 5
Home values in Millersport increase about 4% per year. Mr. Thomas
purchased his home eight years ago for $122,000. What is the
value of his home now?
EXAMPLE 6
The population of a city of one million is increasing at a rate of 3%
per year. If the population continues to grow at this rate, in how
many years will the population have doubled?
EXAMPLE 7
Two different types of bacteria in two different cultures reproduce
exponentially. The first type can be modeled by
B1(t)=1200e0.1532t and the second can be modeled
B2(t)=3000e0.0466t where t is the number of hours. According to
these models, how many hours will it take for the amount of B 1 to
exceed the amount of B2?