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# Exponential equations

## Equations and inequalities: Exponential equations

Exponential equations generally have the unknown variable as the power. The following are examples of exponential equations:

2 x = 1 2 - x 3 x + 1 = 2 4 3 - 6 = 7 x + 2 2 x = 1 2 - x 3 x + 1 = 2 4 3 - 6 = 7 x + 2
(1)

You should already be familiar with exponential notation. Solving exponential equations is simple, if we remember how to apply the laws of exponentials.

### Investigation : Solving Exponential Equations

Solve the following equations by completing the table:

 2 x = 2 2 x = 2 x x -3 -2 -1 0 1 2 3 2 x 2 x
 3 x = 9 3 x = 9 x x -3 -2 -1 0 1 2 3 3 x 3 x
 2 x + 1 = 8 2 x + 1 = 8 x x -3 -2 -1 0 1 2 3 2 x + 1 2 x + 1

### Algebraic Solution

Definition 1: Equality for Exponential Functions

If aa is a positive number such that a>0a>0, (except when a=1a=1


) then:

a x = a y a x = a y
(2)

if and only if:

x = y x = y
(3)
(If a=1a=1, then xx and yy can differ)

This means that if we can write all terms in an equation with the same base, we can solve the exponential equations by equating the indices. For example take the equation 3x+1=93x+1=9. This can be written as:

3 x + 1 = 3 2 . 3 x + 1 = 3 2 .
(4)

Since the bases are equal (to 3), we know that the exponents must also be equal. Therefore we can write:

x + 1 = 2 . x + 1 = 2 .
(5)

This gives:

x = 1 . x = 1 .
(6)

#### Method: Solving Exponential Equations

1. Try to write all terms with the same base.
2. Equate the exponents of the bases of the left and right hand side of the equation.
3. Solve the equation obtained in the previous step.
4. Check the solutions
##### Investigation : Exponential Numbers

Write the following with the same base. The base is the first in the list. For example, in the list 2, 4, 8, the base is two and we can write 4 as 2222.

1. 2,4,8,16,32,64,128,512,1024
2. 3,9,27,81,243
3. 5,25,125,625
4. 13,169
5. 2x2x, 4x24x2, 8x38x3, 49x849x8
##### Exercise 1: Solving Exponential Equations

Solve for xx: 2x=22x=2

###### Solution
1. Step 1. Try to write all terms with the same base. :

All terms are written with the same base.

2 x = 2 1 2 x = 2 1
(7)
2. Step 2. Equate the exponents :
x = 1 x = 1
(8)
LHS = 2 x = 2 1 = 2 RHS = 2 1 = 2 = LHS LHS = 2 x = 2 1 = 2 RHS = 2 1 = 2 = LHS
(9)

Since both sides are equal, the answer is correct.

4. Step 4. Write the final answer :
x = 1 x = 1
(10)

is the solution to 2x=22x=2.

##### Exercise 2: Solving Exponential Equations

Solve:

2 x + 4 = 4 2 x 2 x + 4 = 4 2 x
(11)
###### Solution
1. Step 1. Try to write all terms with the same base. :
2 x + 4 = 4 2 x 2 x + 4 = 2 2 ( 2 x ) 2 x + 4 = 2 4 x 2 x + 4 = 4 2 x 2 x + 4 = 2 2 ( 2 x ) 2 x + 4 = 2 4 x
(12)
2. Step 2. Equate the exponents :
x + 4 = 4 x x + 4 = 4 x
(13)
3. Step 3. Solve for xx :
x + 4 = 4 x x - 4 x = - 4 - 3 x = - 4 x = - 4 - 3 x = 4 3 x + 4 = 4 x x - 4 x = - 4 - 3 x = - 4 x = - 4 - 3 x = 4 3
(14)
LHS = 2 x + 4 = 2 ( 4 3 + 4 ) = 2 16 3 = ( 2 16 ) 1 3 RHS = 4 2 x = 4 2 ( 4 3 ) = 4 8 3 = ( 4 8 ) 1 3 = ( ( 2 2 ) 8 ) 1 3 = ( 2 16 ) 1 3 = LHS LHS = 2 x + 4 = 2 ( 4 3 + 4 ) = 2 16 3 = ( 2 16 ) 1 3 RHS = 4 2 x = 4 2 ( 4 3 ) = 4 8 3 = ( 4 8 ) 1 3 = ( ( 2 2 ) 8 ) 1 3 = ( 2 16 ) 1 3 = LHS
(15)

Since both sides are equal, the answer is correct.

5. Step 5. Write the final answer :
x = 4 3 x = 4 3
(16)

is the solution to 2x+4=42x2x+4=42x.

##### Solving Exponential Equations
1. Solve the following exponential equations:
1. 2x+5=252x+5=25
2. 32x+1=3332x+1=33
3. 52x+2=5352x+2=53
4. 65-x=61265-x=612
5. 64x+1=162x+564x+1=162x+5
6. 125x=5125x=5
2. Solve: 39x-2=2739x-2=27
3. Solve for kk: 81k+2=27k+481k+2=27k+4
4. The growth of an algae in a pond can be modeled by the function f(t)=2tf(t)=2t. Find the value of tt such that f(t)=128f(t)=128

5. Solve for xx: 25(1-2x)=5425(1-2x)=54

6. Solve for xx: 27xÃ—9x-2=127xÃ—9x-2=1


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