# Solving Common Base Exponential Equations Worksheets

When you have an equation that contains a variable that resides as an exponent, we refer to that as an exponential equation. To solve for that variable, you simply reverse engineer the entire equation by countering all of the operations and isolate that variable to one side of the equal sign. In this section we will be focusing on equation that have the same form of base number. It does not have to be the same exact value, but if it is not the same value it will be presented as a constant. Students learn how to solve very common trig. problems with this series of lessons and worksheets.

### Aligned Standard: HSF-LE.A.4

- Find the Variable Step-by-Step Lesson- These problems make it easy to combine, but then you will need to factor your way out of it.
- Guided Lesson - This image always makes me think exponential equations. I was dating a skier in college when I was learning this, I think?
- Guided Lesson Explanation - They follow a simple system here, factor and simplify.
- Practice Worksheet - This is a bit overwhelming at first, but you will get the hang of it.
- Matching Worksheet - Match the x to the equations it goes with.

- Answer Keys - These are for all the unlocked materials above.
- Homework Worksheet 1 - The first problem will give you a challenge, but it is all downhill after that one.
- Homework Worksheet 2 - See how much you know with this one. You should be able to build up and isolate on side of the equation.
- Practice Worksheet 1 - Remember that you can counter positive numbers with negative numbers and vice versa.
- Practice Worksheet 2 - Why not try to settle in with breaking down problems that have different versions of the same variable on both sides.
- Quiz - A good time to check in and see how you are doing with this one.

### Tips for Solving Common Base Exponential Equations

There are two commonly found types of exponential equations. It is all dependent on the base numbers that are presented to you. The base number is value that is being raised to a power by the presence of the exponent. If you have an equation where the same base number appears on both sides or one side possesses a constant, they are pretty easy to work with. That is because we do not have to apply a logarithm to solve them. Whereas to solve exponential equations with unlike bases, one must apply logarithm. Here are some simple rules to remind yourself when you are working to solve problems like these:

**Zero Property** - To solve these equations without using a logarithm, you need to follow several rules. Any variable with zero as its exponent is equal to one. This is known as the zero property. b^{0} = 1

**Negative Property** - If the variable has a negative integer in its exponent, reciprocating it will change the negative exponent into the positive exponent. b^{(-n)} = 1/b^{n}.

**Product Rule** - In multiplication, when the bases are the same, the powers can be added. (b ^{M}) (b^{N})=b^{(M + N)}

**Quotient Rule** - In division, when the bases are the same, the powers can be subtracted. ^{M}/b^{N} =b^{(M - N)}

**Power Rule** - When you raise a base number and its exponent to a new power, you simply need to multiply the exponents to find the final value. (b^{M})^{N} = b^{(M.N)}.

### Solving For That Pesky Exponent

When you identify an exponential equation that has a common base you simply need to set the exponents equal to one another. This can be stated with the form if a^{x} = b^{y}, then x = y. This all holds true as long as the base number is greater than 0 and not equal to one. Time to see an example of this in action: If we were asked to solve for x in:

6^{2x – 1} = 6^{3x + 4}

They both have a common base (6). This means that we can set the exponent to be equal to one another. So, we can rewrite this equation as:

2x – 1 = 3x + 4 (subtract -3x from both sides)

-x -1 = 4 (Add 1 to both sides)

-x = 5 (Divide both sides by -1)

x = -5