# Solving Exponential Equations (Lacking a Common Base)

We have explored how to solve these types of equations when they have the same base, but what happens if the bases are not the same? The first thing you need to do is make sure that you rewrite everything so that a relation can be found between them. This often comes from working backwards from a constant that you started with. These worksheets and lessons help students learn how to approach slightly more complex trig. problems. You will find this type of math very helpful when you are working on understanding an instance of exponential growth or decay.

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

- Solve For a Variable Step-by-Step Lesson- I forgot to tell you that missing value is the exponents.
- Guided Lesson - They lesson gives you a bit of a challenge, but most students look like a deer in headlights when they see number two on here.
- Guided Lesson Explanation - Who would have known that number two is pretty easy when you get it simplified.
- Practice Worksheet - Lots of quick problems. You need about two sheets of scrap paper for these guys.
- Matching Worksheet - Match the values of the exponents to the equations that they fit into.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

The main goal here is to learn how to switch back and forth between the different formats with ease.

- Homework 1 - There are ten problems for you to work through here.
- Homework 2 - Start by evaluating the constant values and what you can transform them into.

### Practice Worksheets

These seem to be part brain teaser for kids. Help them organize their thoughts with these.

- Practice 1 - This is where we can have students start to work on their own with this skill.
- Practice 2 - Notice anything unique about problem number 4?

### Math Skill Quizzes

Remind students to read these problem carefully. The context might trip them up.

- Quiz 1 - This is helpful to gauge where you are at with this skill.
- Quiz 2 - Another helpful assessment tool for you.

### How to Solve Exponential Equations That Lack a Common Base

We all know how to solve basic algebraic equation for separate variables. There are various ways that you can use to solve algebraic equations including completing-squares, middle-term breaking, using the quadratic formula, and factoring. However, things get challenging when in an algebraic equation, the variable is in the exponent. Such equations are what we term as exponential equations. There are two types of exponential equations, one where the bases are same and the second one where the bases are not same. When we are dealing with exponential equations where the bases are not same, we bring log into use. We must first rewrite each side of the equation using the same base. We then often can use the Power of a Power Property where we simply add up all the exponents.

Example: Solve for x, 3^{x} = 11 | You can start solving this by taking logarithm on both sides. As logarithm must have a base, you can choose the base of the exponential equation. In this case, we can take 3 as the base of the logarithmic function.

log_{3 } 3^{x} = log_{3} 11 | xlog_{3 }3 = log_{3} 11 | On solving this equation further, we get x = 2.183.

You can now substitute the value of x into the exponential function to find the final answer.