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5.4 Exponential and Logarithmic
Equations
Essential Questions:
How do we solve
exponential and
logarithmic equations?
Solving Simple Equations
Original
Equation
Rewritten
Equation
Solution
a. 2 x  32
2 x  25
x5
b. ln xx ln3  0
ln x  ln3
x 3
1
c.    9
3
d. e x  7
3 x  32
x  2
ln e x  ln 7
x  ln 7
e. ln x  3
eln x  e3
f. log10 x  1
10 log10 x  10 1
x  e3
x  101 
1
10
Strategies for Solving Exponential and
Logarithmic Equations
1.
2.
3.
Rewrite the given equation in a form that allows the use of the
One-to-One Properties of exponential or logarithmic functions.
Rewrite an exponential equation in logarithmic form and apply
the Inverse Property of logarithmic functions.
Rewrite a logarithmic equation in exponential form and apply
the Inverse Property of exponential functions.
Solving Exponential Equations
Solve each equation and approximate the result to three
decimal places.
x
x
a. e  72
b. 3(2 )  42

Solution:
e  72
ln e x  ln 72
a.
x
x  ln 72
x  4.277
b.
Take natural log of
each side.
Use a calculator.
3(2 x )  42
2 x  14
log 2 2 x  log 2 14
x  log 2 14
x  3.807
Solving an Exponential Equation

Solve e  5  60 and approximate the result to three decimal
places.
x
Solution:
e  5  60
x
e  55
x
ln e x  ln 55
Original equation.
Subtract 5 from each side.
Take natural log of each side.
x  ln 55
Inverse Property.
x  4.007
Use a calculator.
2 n5
Solve
decimal places.
2(3
2 n5
2(3
)  4  11
)  4  11
)  15
15
2 n 5
3

2
2 n5
2(3
2 n5
log 3 3
15
 log 3
2
and approximate the result to three
15
2n  5  log 3
2
2n  5  log3 7.5
5 1
n   log3 7.5
2 2
n  3.417
Solving a Logarithmic Equation
a. Solve ln x = 2
b. Solve
log3 (5x  1)  log3 ( x  7)
Solution:
a.
ln x  2
eln x  e 2
Exponentiate each side.
x  e2
b.
log3 (5x  1)  log3 ( x  7)
5x  1  x  7
4x  8
x2
Check this in the
original equation.
Solving a Logarithmic Equation

Solve 5  2ln x  4 and approximate the result to three
decimal places.
Solution:
5  2ln x  4
2ln x  1
1
ln x  
2
e
ln x
e
1 2
x  e1 2
x  0.607
Exponentiate each
side.
Solving a Logarithmic Equation

Solve
2log 5 3x  4.
Solution:
2log5 3x  4
log5 3x  2
log5 3 x
5
5
2
3x  25
Exponentiate each side (base 5).
25
x
3
Checking for Extraneous Solutions

Solve log10 5 x  log10 ( x  1)  2
Solution:
log10 5 x  log10 ( x  1)  2
log10 [5 x( x  1)]  2
log10 (5 x2 5 x )
10
 10
2
5 x  5 x  100
2
5( x  x  20)  0
2
( x  4)( x  5)  0
x  4 x  5
The solutions appear to be 5
and -4. However, when you
check these in the original
equation, only x = 5 works.
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