# Converting Between Logarithmic and Exponential Forms

The exponential and log forms are direct inverses of one another. Exponential form is used to display that something being repeatedly multiplied by itself in shorthand structure. This feature uses, as the name implies, exponents to achieve this. A logarithm uses the exponent to amplify the base number in order to create that value. Since these formats are directly inverse, we can convert back and forth between them with ease. Students will need to master this skill to understand the nature of situations where there is exponential decay or growth. You can use these lessons and worksheets to learn how to transfer values between exponential and logarithmic forms.

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

- Exponential Form Step-by-Step Lesson- Exponential form from logarithmic form. A simple step to get us started.
- Guided Lesson - The basketball court graphic you ask? I once sat in a lecture about how almost all shots taken in college basketball can be predicted all based on crowd noise. Yes, it was a log calculation.
- Guided Lesson Explanation - It is all based on a basic conversion that most people can do with ease.
- Practice Worksheet - Remind students that you can't turn a log into a log or an exponent into an exponent.
- Matching Worksheet - Match the conversion with the sentence that explains it.
- Express Exponential Forms in Logarithmic Form Worksheet Five Pack - Make them log based.
- Express Logarithmic Form in Exponential Forms Worksheet Five Pack - Time to go all exponential; over the place!

- 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 - Practice both skills wih six problems of each type.
- Homework 2 - You will evaluate the given expression. Make sure to pay attention to the base value.
- Homework 3 - You will expand the logarithms and condense them.

### Practice Worksheets

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

- Practice 1 - Remember that you must satisfy the negative value before you can really get to the meat of the problem.
- Practice 2 - The whole number that follows the base value is called the anti-logarithm.
- Practice 3 - In most cases we will be following the product rule.

### Math Skill Quizzes

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

- Quiz 1 - This may seem like busy work, at this point, but it will pay off big time when working with advanced algebra problems.
- Quiz 2 - Evaluate how anti-logarithm adjusts the overall expression.
- Quiz 3 - We will work with the different rules of expansion to complete these problems.

### How to Convert Between Logarithmic and Exponential Forms

Logarithmic and exponential forms are an important part of mathematics. Given that, they are the core concepts used behind the calculation of the magnitudes of the earthquakes. For example, you can compare the magnitudes of two earthquakes, by converting between logarithmic and exponential form.

For example, the amount of energy released from the first earthquake was 500 times greater than the energy released from the other. The equation that represents this is the problem is 10^{x} = 500, where x is the difference of magnitude placed on the Richter scale. We have to solve for x to understand the force of this energy.

Now, there are different ways to convert them. The first is by using the graphing method. However, estimating from the graphing method can result in an imprecise answer. They give us a general idea based on values that are present and the trends found in them, yet they are not definitive. Thus, we use a log to convert the above exponential function in a logarithmic function. Let's find out practically. They take on the form of the skeleton equation: b^{y} = x. First, we have to learn the values of b, y, and x, to write the equation in log form. The b is the base for log form, while x and y are the unknown variables of the function. In the present example 10^{x} = 500, The value of b is 2, while the value of x and y are 3 and 8, respectively.