# Logarithmic Expressions

Students often come across this topic and are instantly intimidated by the math before. If you simply take your time as a teacher and explain that these are just the inverse of exponents, students will calm down about quickly. I cannot tell you how difficult it is to communicate this to students who have previously learned about this topic without having that pointed out to them. It helps if you equate it to operations that counter one another such as division and multiplication. These types of expressions are helpful for very large or small values. You will need a calculator for several of these sheets. Make sure to have oner handy. This is a collection of worksheets and lessons that teach students to be able to read, interpret, condense and expand logarithmic expressions.

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

- Operational Logs Step-by-Step Lesson- Some students just see the setup of these problems and get automatically intimidated.
- Guided Lesson - I bring you a more complete look at these concepts with this one.
- Guided Lesson Explanation - The last two problems are much easier than they look.
- Practice Worksheet - Three operational based problems, four exponentials to logs, and 3 log solutions for you.
- Matching Worksheet - Looking back at this sheet, I can see how it can overwhelm students a bit.

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

### Homework Sheets

We go over the three skills that relate directly to these types of expressions.

- Homework 1 - The focus here is on expansion.
- Homework 2 - Condense each expression to a single logarithm.
- Homework 3 - Approximate to the nearest hundredth by using a calculator.

### Practice Worksheets

This provides you with more work on the same range of topics.

- Practice 1 - The power rule comes in really helpful here at times.
- Practice 2 - Always start by looking at the coefficients and see if there is any wiggle room in there.
- Practice 3 - These are consuming calculations that generate very large or small values.

### Skill Quizzes

It is always a good idea to see where you are at with a skill.

- Quiz 1 - This will require you to run a basic series of operations.
- Quiz 2 - This quiz basically requires that you reverse engineer what you did on the previous quiz.
- Quiz 3 - Your first move should be to reshape the expression to suit your purpose.

### What Are Logarithmic Expressions?

Let's take a sneak-peek into logarithms because they are probably a bit difficult for most of the students. In its simplest form, logarithms are exponents, or you may say they are the opposite of exponentials. This is very similar to how the operation of division is the inverse of multiplication and subtraction is the opposite of addition. Similarly, logs are the inverse of exponents. They undo or counter what exponents do.

A good way to demonstrate this is to look at the equation; y = b_{x}. This is equivalent to or means the exact same as log_{b} (y) = x. This is the relationship between the log and the exponent. This is pronounced as log base-b of y is equal to x. In logarithmic expressions, the base b became the exponent of y and multiplied the number of times of the exponent. This is the answer to the logarithmic expression. Log expressions come with rules and can be used to simplify or expand expressions. Since the log expression has too much in it, it is necessary to expand the expression in order to solve it. There is somewhat of a spiral relationship that exists between these values meaning that base and exponent are in a constant flow.

The general standard of logarithmic expressions is often stated as logarithm base (b) of a positive number (x) is such that for any base greater than zero, but not equal to one can be stated as log_{b} x = y if and only is b^{y} = x.

When a base is not indicated in a logarithm it is implied that it has a base of 10. For example, log a = log_{10} a. Logarithms that have a base of 10 are referred to as common. Logarithms that have a base of the mathematical constant e are referred to as natural logarithms. As you begin to work on the nature of these types of expressions you will get comfortable with the different directions that each will push you in.