1. Solving
Exponential
Equations

2. Question from last day:B. C. D.

3. Question: a. Determine the range of this function.
The graph above is translated 2 units up. What is the new
range?

4. Question

5. Question

6. Learning Outcomes:
To determine the solution of exponential equations in which the bases are powers of one another
To solve problems that involve exponential growth or decay
To solve problems that involve the application of exponential equations to loans, mortgages, and investments.

7. Exponent Laws: Product Law: Quotient Law: Power of a Power:
Power of a Product:
Power of a Quotient:
Integral Exponent Rule:
Rational Exponents:

8. Converting Bases:
Convert 9² to base 3:
Convert to base 2:

9. Type 1: Solving Exponential Equations with a Common Base
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.
Use the following procedure to solve an equation where the variable is in the exponent:
Write both sides of the equation in the same base
Equate the exponents on both sides of the equation
Determine the value of the variable

10. Solving Exponential Equations Examples
Solve the following:
a) x - 3 = 27x + 4
b) 16 2x + 4 = 1
(32)2x - 3 = (33)x + 4
34x - 6 = 33x + 12
16 2x + 4 = 160
2x + 4 = 0
2x = -4
x = -2
Since both sides have the same
base, then the exponents must
be equal:
4x - 6 = 3x + 12
x = 18
Try this:

11. Solving Exponential Equations
(32)x + 2 = (3-3)x + 2
32x + 4 = 3-3x - 6
2x + 4 = -3x - 6
5x = - 10
x = -2
x2 = 5x - 4
x2- 5x + 4 = 0
(x - 4)(x - 1) = 0
x - 4 = 0 or x - 1 = 0
x = 4 or x = 1
-3x + 6 = 4x + 13
-7x = 7
x = -1

12. Exponential Equations with Rational Exponents

13. Type 2: Solving Missing Bases
Solve:
This equation involves the base as the variable.
Isolate the power.
Take the root of the exponent to solve for the missing base.
Take the cube root to isolate the base.

14. Try this: Raise both sides to the power of -4.
What is the key point to solving this type of equation?

15. Graphical Solutions Solve 12=10(1.04)x
This cannot be solved with the methods we have so…
use technology
Y1 = 1.2
Y2 = (1.04)x
Solution is the x value of the intersection point.
x = 4.65

16. Assignment: Handout #1a,b, 2c,d, 3a,c, 4a,b,c,e, 5,7
P. 364 # 2, 3a, 5d, 7b,d, f