Solving Exponentials

By Graphing and Common Base

To Solve Exponential by Graphing

• Enter left side of equation into y1

• Enter right side of equation into y2

• Set window to see interesection• 2nd, trace, 5, enter, enter, enter

Note: Inequalities work the same, except now you are looking to see if y1 is greater/less than y2

Example 1

2x = 8

2nd, trace, 5, enter, enter, enter

y1 y2

X=3

Example 2

4x+1 = 82x+3

2nd, trace, 5, enter, enter, enter

y1 y2

X=-1.75

Example 3

8x+2 < 32

2nd, trace, 5, enter, enter, enter

y1 y2

X< -1/3

Example 4

23x+2 ≥ -2

2nd, trace, 5, enter, enter, enter

y1 y2

X ≥ -2

To Solve Exponential by Common Base

• Make sure that both sides of equal sign share a common base.

• Set the exponents equal to each other and solve.

Example 1

5x = 53

-They both share a common base.-Set both exponents equal to each other and solve.

x = 3

Base of 5

Base of 5

Example 2

101-x = 104

-They both share a common base.-Set both exponents equal to each other and solve.

Base of 10

Base of 10

1-x = 4 -1 -1 -x = 3 x = 3

Example 3

76x = 72x-20

-They both share a common base.-Set both exponents equal to each other and solve.

Base of 7

Base of 7

6x = 2x-20-2x -2x4x = -204 4

x = -5

Example 4

92x = 27x-1

-They do NOT share a common base so we have to change the bases to be alike.-Set both exponents equal to each other and solve.

Base of 9

Base of 27

(32)2x = (33)x-1

2(2x) = 3(x-1) 4x = 3x – 3 -3x -3x x = -3

Base of 3

Base of 3

You Try Example 5

22x = 82x-4

-They do NOT share a common base so we have to change the bases to be alike.-then Set both exponents equal to each other and solve.

Base of 2

Base of 8

(2)2x = (23)2x-4

2x = 3(2x-4)2x = 6x-12-6x -6x-4x = -12-4 -4 x = 3

Base of 2

Base of 2

Example 6

9x-4 =

-They do NOT share a common base so we have to change the bases to be alike.-then Set both exponents equal to each other and solve.

Base of 9

Base of 1/81

9x-4 = (9-2)

x-4 = -2 +4 +4 x = 2

Base of 9

Base of 9

1

81

Example 7

813-x =

-They do NOT share a common base so we have to change the bases to be alike.-then Set both exponents equal to each other and solve.

Base of 81

Base of 1/3

(34)(3-x)= (3-1)5x-6

4(3-x) = -1(5x-6) 12-4x= -5x+6 12+x=6 x = -6

Base of 3

Base of 3

5 61

3

x

Example 8

2510x+8= 4-2x

-They do NOT share a common base so we have to change the bases to be alike.-then Set both exponents equal to each other and solve.

Base of 25

Base of 1/125

(52)10x+8=(5-3)4-2x

2(10x+8) = -3(4-2x) 20x+16 = -12+6x 14x+16 = -12 14x = -28 x = -2

Base of 5

Base of 5

1

125