EXPONENTIAL AND
LOGARITHMIC EQUATIONS
Section 3.4
EXPONENTIAL & LOG EQUATIONS
 In
a)
b)
c)

the previous sections, we covered:
Definitions of logs and exponential functions
Graphs of logs and exponential functions
Properties of logs and exponential functions
In this section, , we are going to study procedures
for solving equations involving logs and
exponential equations
e.g. :
e 2x  9e x  36  0
EXPONENTIAL & LOG EQUATIONS
 In
the last section, we covered two basic properties,
which will be key in solving exponential and log
equations.
1.
One-to-One Properties
a) a  a  x  y
x
2.
y
b) log a x  log a y  x  y
Inverse Properties
a) log a a x  x
b) a
loga x
x
EXPONENTIAL & LOG EQUATIONS
 We
can use these properties to solve simple
equations:
2 x  32
x
2 2
5
x
2
x
1
  9
3
ln x   3
3 3
 e ln x  e 3
 x 5
 x -2
 x  e -3
EXPONENTIAL & LOG EQUATIONS
 When
solving exponential equations, there are two
general keys to getting the right answer:
1.
Isolate the exponential expression
2.
Use the 2nd one-to-one property
log a x  log a y  x  y
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation:
4 x  72
Isolate the exponential expression:
Apply the 2nd one-to-one property
log 4 4 x  log 4 72
log 72
 3.085...
x 
log 4
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation:
3(2 )  42
x
Isolate the exponential expression:
2 x  14
Apply the 2nd one-to-one property
log 2 2 x  log 214
log
14
 3.807...
x 
log 2
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation:
4(e )  16
2x
Isolate the exponential expression:
e
2x
4
Apply the 2nd one-to-one property
ln e  ln 4
2x
2x  ln 4
ln 4
x 
 0.693
2
EXPONENTIAL & LOG EQUATIONS
 Solve
a)
b)
c)
the following equations:
e x  5  60
 x  ln 55  x  4.007
ln 4
2(3  1)  10  x 
ln 3
x
5(e
x2
 x  1.262
22
- 2  x  - 0.518
)  8  14  x  ln
5
EXPONENTIAL & LOG EQUATIONS
 Solving
Equations of the Quadratic Type

Two or more exponential expressions

Similar procedure to what we have been doing

Algebra is more complicated
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation:
e 2 x  3e x  2  0
Start by rewriting the equation in quadratic form.
(e )  3e  2  0
x 2
x
Factor the quadratic equation:
let x  e x  x 2  3 x  2  0  ( x  2)( x  1)  0
(e x  2)(e x  1)  0
EXPONENTIAL & LOG EQUATIONS
(e  2)(e  1)  0
x
x
e 2 0
ex 1  0
e 2
ex  1
x
x
ln
ex
 ln 2
x  ln 2
x  .693
x0
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation:
e 2 x  e x  20  0
(e )  e  20  0
x 2
x
e 5  0
x
e 5
x  ln 5
x  1.609
x
 (e x  5)(e x  4)  0
e 40
x
e  4
x  ln 4
x  error
x
EXPONENTIAL AND
LOGARITHMIC EQUATIONS
Section 3.4
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation:
8  3x  5
5
3 
8
x
x
5
 log 3 3  log 3
8
x
log 5
8
log 3
 x  0.428
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation:
5 x  2  33 x  2
Since these are exponential functions of a different
base, start by taking the log of both sides
log 5 x 2  log 33 x  2
( x  2) log 5  (3x  2) log 3
EXPONENTIAL & LOG EQUATIONS
( x  2) log 5  (3x  2) log 3
x log 5  2 log 5  3 x log 3  2 log 3
x log 5  3x log 3  2 log 3  2 log 5
x(log 5  3 log 3)  2 log 3  2 log 5
2 log 3  2 log 5
x
 3.212
log 5  3 log 3
EXPONENTIAL & LOG EQUATIONS
 So
far, we have solved only exponential equations
 Today,
we are going to study solving logarithmic
equations
 Similar
to solving exponential equations
EXPONENTIAL & LOG EQUATIONS
 Just
as with exponential equations, there are two
basic ways to solve logarithmic equations
1)
Isolate the logarithmic expression and then write
the equation in equivalent exponential form
2)
Get a single logarithmic expression with the same
base on each side of the equation; then use the
one-to-one property
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation:
ln x  2
Isolate the log expression:
Rewrite the expression in its equivalent exponential form
e x
2
x  7.389
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation.
log 3 (5x  1)  log 3 ( x  7)  0
Get a single log expression with the same base on each
side of the equation, then use the one-to-one property
log 3 (5 x  1)  log 3 ( x  7)
5x  1  x  7
x2
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation
2 log 5 3x  4
Isolate the log expression:
log 5 3x  2
Rewrite the expression in exponential form
5  3x
25
x
3
2
EXPONENTIAL & LOG EQUATIONS
 In
some problems, the answer you get may not be
defined.
 Remember,
 Therefore,
y  log a x is only defined for x > 0
if you get an answer that would give you
a negative “x”, the answer is considered an
extraneous solution
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation
log 10 5x  log 10 ( x  1)  2
Isolate the log expression:
log 10[5x( x  1)]  2
Rewrite the expression in exponential form
10  5 x( x  1)
2
EXPONENTIAL & LOG EQUATIONS
100  5 x  5 x
2
x  x  20  0
2
( x  5)( x  4)  0
x   4, 5
Would either of these give us an undefined logarithm?
log 10 5x  log 10 ( x  1)  2
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equations:
 x  10
5
 x  316.228
a)
2 log x  5
b)
ln x  2  ln x  x  1, 2
c)
log x  log( x  3)  1
2
10
x
3
EXPONENTIAL AND
LOGARITHMIC EQUATIONS
Section 3.4 - Applications
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation:
2 log 5 x  log 5 9
log 5 x 2  log 5 9
x2  9
x3
x  3
EXPONENTIAL & LOG EQUATIONS
 Solve
the following equation:
log 4 ( x  3)  log 4 (2  x)  1
log 4 ( x  3)(2  x)  1
4  ( x  3)( 2  x)
x 2  x  6  4
x  x20
2
EXPONENTIAL & LOG EQUATIONS
x2  x  2  0
x2  x  2  0
EXPONENTIAL & LOG EQUATIONS
 How
long would it take for an investment to double
if the interest were compounded continuously at
8%?
What is the formula for continuously compounding interest?
A  Pert
If you want the investment to double, what would A be?
A  2P
EXPONENTIAL & LOG EQUATIONS
A  Pert
2 P  Pe0.08t  2  e 0.08t
 0.08t  ln 2
 ln 2  ln e
ln 2
t 
0.08
It will take about 8.66 years to double.
0.08t
EXPONENTIAL & LOG EQUATIONS
 You
have deposited $500 in an account that pays
6.75% interest, compounded continuously. How
long will it take your money to double?
EXPONENTIAL & LOG EQUATIONS
 You
have $50,000 to invest. You need to have
$350,000 to retire in thirty years. At what
continuously compounded interest rate would you
need to invest to reach your goal?
EXPONENTIAL & LOG EQUATIONS
 For
selected years from 1980 to 2000, the average
salary for secondary teachers y (in thousands of
dollars) for the year t can be modeled by the
equation:
y = -38.8 + 23.7 ln t
Where t = 10 represents 1980. During which year did
the average salary for teachers reach 2.5 times its
1980 level of $16.5 thousand?
EXPONENTIAL & LOG EQUATIONS
EXPONENTIAL & LOG EQUATIONS
EXPONENTIAL & LOG EQUATIONS