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• How do I solve exponential equations and inequalities?
4.6 Solve Exponential Equations and Inequalities
Example 1 Solve by equating exponentsSolve by equating exponents
.1664 Solve 1 xx
Write original equation.11664 xx
Rewrite each power with a base __. 1______ xx4
34 24Power of a power property.__________ x34 224 x
Property of equality._________ x3 22 xSolve for x.___x 2
The solution is ___.2Check Substitute the solution into the original equation.
1____ 1664 2 2
________ 4096 4096
4.6 Solve Exponential Equations and Inequalities
Checkpoint. Complete the following exercises.Checkpoint. Complete the following exercises.
.93 Solve 1. 237 xx
337 x
1x
23x2
373 x x43xx 437
x3 3
4.6 Solve Exponential Equations and Inequalities
Example 2 Solve an exponential equation by graphingSolve an exponential equation by graphing
.4
12 Solve
3
xx
3
4
1 and 2Graph
xx yy
The graphs intersect only once, when x = ___. 2
So, ___ is the only solution.2
plane. coordinate same in the
4.6 Solve Exponential Equations and Inequalities
Checkpoint. Complete the following exercises.Checkpoint. Complete the following exercises.
.2
14 Solve 2.
1
xx
22x
3
1x
121x
x22 12 x
12 xxx3 1
4.6 Solve Exponential Equations and Inequalities
Example 3 Solve an exponential inequalitySolve an exponential inequality
.93 Solve 34 zz
Write original equation.34 93 zz
Rewrite each power with a base __. 34 _____ zz33 23
Power of a power property.__________ z43 623 z
Because f( x ) = 3x is an increasing function, f( x1 ) < f( x2 ) implies that x1 < x2.
_________ z4 62 z
Solve for x.___z 3
4.6 Solve Exponential Equations and Inequalities
Example 3 Solve an exponential inequalitySolve an exponential inequality
.93 Solve 34 zz
Solve for x.___z 3
Substitute z = 2.
Check Check that the solution is reasonable by substituting several values into the original inequality.
32?
24 93 ____________6561 049,59
Substitute z = 4.
34?
44 93 ___________________721,046,43 969,782,4
Because z = __ is a solution of the inequality and z = __ is not,
the solution z < ___ is reasonable.
2 43
4.6 Solve Exponential Equations and Inequalities
Checkpoint. Complete the following exercises.Checkpoint. Complete the following exercises.
.42 Solve 3. 121 xx
21x
1x
2212 x
12 x 242 x
241 xx3 x3
4.6 Solve Exponential Equations and Inequalities
Example 4 Solve an exponential inequality by graphingSolve an exponential inequality by graphing
.75.033 Solve 12 x
75.0 and 33Graph 12 yy x
The graphs intersect only once, when x = ______. 1.10
The graph of y = 32x1 – 3 is ______ the graph of y = 0.75 when _______
above
plane. coordinate same in the
x > 1.10
The solution is _________.x > 1.10
4.6 Solve Exponential Equations and Inequalities
Checkpoint. Complete the following exercises.Checkpoint. Complete the following exercises.
.5.042 Solve 4. 1 x
17.3x17.3x
4.6 Solve Exponential Equations and Inequalities
Pg. 144, 4.6 #1-19
