SECTION 10.2 Exponential Functions 583
Solving an Exponential Equation
Step 1 Each side must have the same base.If the two sides of the equa-
tion do not have the same base, express each as a power of the same
base if possible.
Step 2 Simplify exponentsif necessary, using the rules of exponents.
Step 3 Set exponents equalusing the property given in this section.
Step 4 Solvethe equation obtained in Step 3.Solving an Exponential EquationSolve the equation
Write with the same base;
and. (Step 1)
Power rule for exponents (Step 2)
If then. (Step 3)Solve for x. (Step 4)CHECK Substitute for x.
✓ True
The solution set is E^32 F. NOW TRY
9 x= 9 3/2 = 19 1/2 23 = 33 = 27
3
2x=
3
2
2 x= 3 ax=ay, x=y
32 x = 33
9 = 32 27 = 331322 x = 33
9 x = 27
9 x=27.
EXAMPLE 4
Solving Exponential EquationsSolve each equation.
(a)
Write with the same base;
Power rule for exponents
Set exponents equal.x= 5 Subtract 2x. Add 1.
3 x- 1 = 2 x+ 4
43 x-^1 = 42 x+^4
4 3 x-^1 = 1422 x+^216 = 42.
43 x-^1 = 16 x+^2
EXAMPLE 5
Be careful multiplying
the exponents.Verify that the solution set is
(b)
Write with the same base;
Set exponents equal.CHECK ✓ Substitute for x; true
The solution set is 5 - 36.
6 x = 6 -^3 = - 3
1
63
=
1
216
x=- 3
6 x = 6 -^3613 = 6 -^3.
6 x = 216 = 63
1
63
6 x=
1
216
556.
NOW TRY
EXERCISE 4
Solve the equation.
8 x= 16NOW TRY ANSWER
- E^43 F