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Contents
Exponential Equations 1 2
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Exponential Equations
An exponential equations is one
that contains a variable with an index.
To solve exponential equations, we
try to reduce it into the simplest form:
ax = ay,
 x = y
condition: a cannot take the
values -1, 0 or 1.
Solve
the following equations:
5x = 1/125
= 5-3
x = -3
16x = 32
24x = 25
4x = 5
x = 1.2
Substitution Method
Using y = 3x, solve the following equation:
3 2x + 1
+ 2 (3x) =
1
Solution:
(Rearrange the equation such
that it is in terms of 3x)
3 (3x)2
+ 2 (3x)
= 1
Substituting y = 3x
,
3y2 + 2y = 1
(Solve as in a quadratic
equation)
3y2 + 2y - 1 = 0
(3y - 1) (y + 1) = 0
y = 1/3 or y = -1
Substituting into y =
3x
,
3x = -1 no
real solution as 3x > 0
3x
= 1/3
x = -1
Example:
 Solve
:
22x+3 + 2x+4 = 2 + 2x
Since the substitution is not given, we have to use our own substitution. By
observation, it seems to be that 2x has to be substituted. Rearrange
the equation:
23 (2x)2
+ 24 (2x)
= 2 + 2x
Let y = 2x,
8y2 + 16y = 2 + y
8y2 + 15y - 2 = 0
(8y - 1) (y + 2) = 0
y = 1/8 or y = -2
Substituting into y =
2x,
2x =
-2 no real
solution as 2x > 0
2x
= 1/8
x = -3
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