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(3 2 ) x + 2 = (3 -3 ) x x + 4 = 3 -3x - 6 2x + 4 = -3x - 6 5x = - 10 x = -2 x 2 = 5x - 4 x 2 - 5x + 4 = 0 (x - 4)(x - 1) = 0 x - 4 = 0 or x - 1 = 0 x = 4 or x = Solving Exponential Equations -3x + 6 = 4x x = 7 x = -1
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2.4.1MATHPOWERTM 12, WESTERN EDITION
2.4Chapter 2 Exponents and Logarithms
To solve exponential equations, you need to apply the Laws of Exponents. One method of solving exponential equations is based on the following property:
If ax = ay, then x = y.
That is, if 2 powers are equal and have the same bases, then the exponents are equal.Solve the following:
a) 92x - 3 = 27x + 4
(32)2x - 3 = (33)x + 4
34x - 6 = 33x + 12
Since both sides have the samebase, then the exponents mustbe equal:
4x - 6 = 3x + 12 x = 18
b) 16 2x + 4 = 116 2x + 4 = 160
2x + 4 = 0 2x = -4 x = -2
2.4.2
Solving Exponential Equations
c) 9x 2
127
x2
(32)x + 2 = (3-3)x + 2
32x + 4 = 3-3x - 6
2x + 4 = -3x - 6 5x = - 10 x = -2
d) 2x 2
(16x 1 ) 2x
2x2
(24 x 4) 2x
2x2
25 x 4
x2 = 5x - 4 x2- 5x + 4 = 0 (x - 4)(x - 1) = 0 x - 4 = 0 or x - 1 = 0 x = 4 or x = 1
2.4.3
Solving Exponential Equations
e) 1
8
x 2
2 16x 3
2 3 x 221 24 x 3
2 3x 6 21 24 x 12
2 3x 6 24x 13
-3x + 6 = 4x + 13 -7x = 7 x = -1
Exponential Growth
A cell doubles every 4 min. If there are 500 cells originally, how long would it take to reach 16 000 cells?
N(t ) No 2td
N(t) Number of bacteria after t minutes
No Number of bacteria originally t Time passedd Doubling time
N(t ) No 2td
16 000 500 2t4
32 2t4
25 2t4
5 t4
t = 20
Therefore, it would take20 min for the cells toreach 16 000. 2.4.4
Exponential Decay
The half-life of sodium 24 is 15 h. How long would it takefor 1600 mg to decay to 100 mg?
A(t) Ao 12
th
A(t) Amount after a given period of time
Ao Amount originally t Time passedh Half-life A(t) Ao
12
th
100 160012
t15
116
12
t15
12
4
12
t15
4 t
15t = 60
It would take 60 hto decay to 100 mg.
2.4.5
Suggested Questions:Pages 89 and 901-16, 24, 30,32, 34 a
Page 93all
2.4.6
