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Name _________________________________________________________ Date __________
6.6 Solving Exponential Equations
For use with Exploration 6.6
Essential Question How can you solve an exponential equation
graphically?
1 EXPLORATION: Solving an Exponential Equation Graphically
Go to BigIdeasMath.com for an interactive tool to investigate this exploration.
Work with a partner. Use a graphing calculator to solve the exponential equation
2.5x−3 = 6.25 graphically. Describe your process and explain how you determined
the solution.
2 EXPLORATION: The Number of Solutions of an Exponential Equation
Go to BigIdeasMath.com for an interactive tool to investigate this exploration.
Work with a partner.
x
y = 2.
a. Use a graphing calculator to graph the equation
6
−6 6
−2
b. In the same viewing window, graph a linear equation (if possible) that does not
x
y = 2.
intersect the graph of
c. In the same viewing window, graph a linear equation (if possible) that intersects
the graph of y = 2x in more than one point.
d. Is it possible for an exponential equation to have no solution? more than one
solution? Explain your reasoning.
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6.6 Solving Exponential Equations (continued)
3 EXPLORATION: Solving Exponential Equations Graphically
Go to BigIdeasMath.com for an interactive tool to investigate this exploration.
Work with a partner. Use a graphing calculator to solve each equation.
2x = 1 x+1 x
a. b. 20= c.
2 21=
x x−1 42x = 1
d. 39= e. 30= f. 16
3x 1 x+2 1 x−2 3
2 = 3 = 22=−x
g. 8 h. 9 i. 2
Communicate Your Answer
4. How can you solve an exponential equation graphically?
5. A population of 30 mice is expected to double each year. The number p of mice in
the population each year is given by p = 30 2n . In how many years will there be
()
960 mice in the population?
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6.6 Practice
For use after Lesson 6.6
Core Concepts
Property of Equality for Exponential Equations
b ≠ 1,
Words Two powers with the same positive base b, where are equal if
and only if their exponents are equal.
x 5 x = 5, x 5
Numbers If then x = 5. If then 2 = 2.
2 = 2,
b ≠ 1, x y
Algebra If and then bb= if and only if x = y.
b > 0
Notes:
Worked-Out Examples
Example #1
Solve the equation. Check your solution.
9x 7x+8 9x 7x+8
3 = 3 Check: 3 = 3
9(4) ? 7(4)+8
9x = 7x + 8 3 = 3
?
36 28+8
− 7x − 7x 3 = 3
36 36
2x = 8 3 = 3
2x 8 17 17
—
1.50095 × 10 = 1.50095 × 10 ✓
=
— 2
2
x = 4
Th e solution is x = 4.
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6.6 PracticePRACTI(CcoEnytinued)
Example #2
Solve the equation. Check your solution.
1 x+1
−3x+3 —
36 =
)
(216 Check:
1 x+1 x+1
2 −3x+3 — 1
(6) =
) −3x+3
( 3 —
36 =
)
6 (216
2(−3x+3) −3 x+1
6 = (6 ) −3(3)+3? 1 3+1
36 =
2(−3x+3) −3(x+1) —
6 = 6 (216)
2(−3x + 3) = −3(x + 1) −9+3 ? 1 4
36 =
—
2(−3x) + 2(3) = −3(x) − 3(1) (216)
−6 ? 1
36 =
−6x + 6 = −3x − 3 —
2164
+ 6x + 6x ?
1 1
=
— ——
6 = 3x − 3 366 2,176,782,336
+ 3 + 3 1 1
——
✓
=
9 = 3x ——2,176,782,336
2,176,782,336
9 3x
—
=
— 3
3
3 = x
Th e solution is x = 3.
Practice A
Extra Practice
In Exercises 1–15, solve the equation. Check your solution.
4x 12 x+5 20 45x− 2x
1. 33= 2. 88= 3. 66=
6x−3 −+34x 2x+11 3−2x
4. 55= 5. 4 = 1024 6. 8 = 512
7−x x−2 6x−1 5x
7. 4 = 256 8. 49 = 343 9. 36 = 6
x−43x x+1 x 2x 21x+
10. 9 = 81 11. 64 = 512 12. 6 = 36
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