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Section 3.4 Solving Exponential and Logarithmic Equations 225
Section 3.4
Solving Exponential and Logarithmic Equations
■
To solve an exponential equation, isolate the exponential expression, then take the logarithm of both sides.
Then solve for the variable.
1. log a a x x
2. ln e x x
■ To solve a logarithmic equation, rewrite it in exponential form. Then solve for the variable.
1. a log a x x
2. e ln x x
■ If a > 0 and a 1 we have the following:
1. log a x log a y ⇒ x y
2. a x a y ⇒ x y
■ Use your graphing utility to approximate solutions.
Vocabulary Check
1. solve 2. (a) x y (b) x y (c) x (d) x
3. extraneous
1.
4 2x7 64 2.
(a) x 5
4 257 4 3 64
2 3x1 32
(a) x 1
2 311 2 2 1 4
Yes, x 5 is a solution.
No, x 1 is not a solution.
(b)
x 2
(b)
x 2
4 227 4 3 1 64 64
2 321 2 7 128
No, x 2 is not a solution.
No, x 2 is not a solution.
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3.
3e x2 75
(a) x 2 e 25
3e 2e25 + 2 3e e25 75
No, x 2 e 25 is not a solution.
(b) x 2 ln 25
3e 2ln 252 3e ln 25 325 75
Yes, x 2 ln 25 is a solution.
(c) x 1.2189
3e 1.21892 3e 3.2189 75
4.
4e x1 60
(a) x 1 ln 15
4e 1ln 151 4e ln 15 415 60
Yes
(b) x 3.7081
4e 3.70811 4e 2.7081 60
Yes
(c) x ln 16
4e ln 161 23.5 60
Yes, x 1.2189 is a solution.
No