## How to solve Exponents

## How to solve

To solve exponential equations without logarithms, you need to have equations with comparable exponential expressions on either side of the "equals" sign, so you can compare the powers and solve. In other words, you have to have "(some base) to (some power) equals (the same base) to (some other power)", where you set the two powers equal to each other, and solve the resulting equation. For example:

**Solve 5***x*= 5**3****.**

- Since the bases ("5" in each case) are the same, then the only way the two expressions could be equal is for the powers also to be the same. That is:

*x*= 3**Solve 101–***x*= 104

- Since the bases are the same, I can equate the powers and solve:

- 1 –

*x*= 4

1 – 4 =

*x*

**–3 =**

*x***Solve 3**Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved*x*= 9.

- Since 9 = 32, this is really asking me to solve:

- 3

*x*= 32

*x*= 2**Solve 32***x*–1 = 27.

- In this case, I have an exponential on one side of the "equals" and a number on the other. I can solve the equation if I can express the "27" as a power of 3. Since 27 = 33, then I can convert and proceed with the solution:

- 32

*x*–1 = 27

32

*x*–1 = 33 2

*x*– 1 = 3

2

*x*= 4

*x*= 2*every*value in your calculator can waste a lot of time. You'll want to have a certain degree of facility, of familiarity and speed, by the time you reach the test, so familiarize yourself with the smaller powers now.**Solve 3***x***^2****–3***x*= 81.

- Formatting note: HTML doesn't generally "like" nested superscripts, so the above uses the "carat" notation to denote the exponent.

This exercise works just like the previous one:

- 3

*x*^2–3

*x*= 81

3

*x*^2–3

*x*= 34

*x*2 – 3

*x*= 4

*x*2 – 3

*x*– 4 = 0

(

*x*– 4)(

*x*+ 1) = 0

*x*= –1, 4**Solve 42***x***^2****+2***x*= 8.

- This equation is similar to the previous two but is not quite the same, because 8 is not a power of 4. However, both 8 and 4 are powers of 2, so I can convert:

- 4 = 22 8 = 23 42

*x^*2+2

*x*= (22)2

*x^*2+2

*x*= 2(2)(2

*x*^2+2

*x*) = 24

*x*^2+4

*x*

- 42

*x*^2+2

*x*= 8

24

*x*^2+4

*x*= 23 4

*x*2 + 4

*x*= 3

4

*x*2 + 4

*x*– 3 = 0

(2

*x*– 1)(2

*x*+ 3) = 0

*x*= 1/2 , –3/2**Solve 4***x*+1 = 1/64.

- Negative exponents can be used to indicate that the base belongs on the other side of the fraction line. Since 64 = 43, then I can use negative exponents to convert the fraction to an exponential expression: 1/64 = (43)–1 = 4–3. Using this, I can solve the equation:

- 4

*x*+1 = 1/64 4

*x*+1 = 4–3

*x*+ 1 = –3

*x*= –4**Solve****8***x***–****2****=***sqrt***[8]**

- I need to recall that square roots are the same as one-half powers, and convert the radical to exponential form. Then I can solve the equation:

- 8

*x*–2 = sqrt[8]

8

*x*–2 = 8 1/2

*x*– 2 = 1/2

**x****= 2**

**1**

**/**

**2**

**= 5/2**

**Solve****2****x****= –4**Think about it: What power on the

*positive*number "2" could

*possibly*yield a

*negative*number? A number can never go from positive to negative by taking powers; I can never turn a positive two into a negative

*anything*, four or otherwise, by multiplying two by itself, regardless of the number of times I do the multiplication. Exponentiation simply doesn't work that way. So the answer here is:

**no solution**