# Logarithm Worksheets

What are Logarithms? Logarithms are basically mathematical operations that usually determine how frequent a certain number, normally called a base, gets multiplied by itself to another number. Logarithms usually related geometric expressions into arithmetic progressions. Some of the examples are throughout art and nature like mineral hardness, spacing found on the guitar frets, sound intensities, windstorms, stars, acids and earthquakes. Logarithms also describe how humans are able to instinctively able to think about numbers. Logarithms usually characterize how frequently you need a sheet of paper to be folded to get 64 layers. Each time the paper is folder in half the number of layers is doubled. Speaking mathematically, when 2 is multiplied by a certain number of times you get 64. So how many multiplications are necessary? The easier way of thinking of logarithm is to consider them inverse of an exponential. So, the above equation can have the same meaning: 2x = 64 and since 2 x 2 x 2 x 2 x 2 x 2 = 64, 26 = 64. This means that if we fold a paper 6 times, we will get around 64 layers. Logarithms go hand and hand with exponents and functions. Yet the skill is drastically neglected in the core curriculum. It is a bit baffling to me. Many of the advanced High School courses include logarithms. It seems that the curriculum counsel needs to look at that in the future.

- Calculate the Value of Log Operations Problems - You will evaluate logs that are undergoing all types of operations.
- Converting Between Logarithmic and Exponential Forms - This is a very common thing done in most science labs around the world.
- Converting Between Logarithmic and Exponential Functions - We show you a few tricks that you could use along the way.
- Express as a Single Logarithm - This is a way to simplify the way a value looks.
- Graphing Exponential and Logarithmic Functions - A great way to bring these to like with some great imagery.
- Logarithmic Equations - You can employ some algebra skills here to find unknowns.
- Logarithmic Expressions - You will be asked to expand and evaluate expressions that are in logarithm form.
- Logarithmic Functions - You will be asked to convert values to exponential form, determine ranges, and condense values.
- Logarithmic Graph Paper - This may come in handy. Scroll towards the bottom.
- Logarithm Word Problems - The applications to the financial world here are interesting.
- Multiplying by Powers of 10 - This will put you in the right frame of mind for working with these operators.
- Powers of Ten - Students learn how to express values using this form of notation as well as understand values that are in its place.
- Solve for the Unknown (Using Logarithms) Problems - That algebra thing comes in very handy here.

### Why Is It Important to Understand Logarithms?

Essentially, they are a quick way to communicate large numbers since they work off of the base ten system. This can be helpful when you are analyzing or discussing exceptionally large objects like planets or stars. Another profound use is that they are the inverse function to exponential functions. This will save you tons of time and give you other options when working with most differential equations. Since adding and subtracting logarithms is equal to multiplication and division, it can help us work through many complex math situations. In the real world, we often use this format to express the concept of decay whether it be in the sense of cooling, heating, or calculating frictional forces. As you move into advanced math and science, this form of notation and operational math will be invaluable and save you tons of time along the way.

### How Are Logarithms Used in the Real World?

Learning math is much more rewarding when we consider the value of what it can help us do. When ever we describe a value in terms of powers of ten, we are describing things in logarithmic means. This applies to countless areas of the real world. Have you ever used phrases that discuss things in this nature? You probably didn't even realize it. When we use terms such as "triple digits", "seven figures", "inflation rate", or "interest rate" we are discussing things in this manner. Logarithms take the concept of multiplication as a step to increase or decrease. All the while they are looking at things in terms of the powers of 10. Either 10 times greater or lesser. Let's take a look at some of the walks of life that we use this concept:

**Financial World Uses** - Logarithms are used to determine how we pay back loans and how we receive our salary. Almost all financial vehicles that are used to stage investment growth or decay use logarithmic based means. They are your best friend when you're collecting the money, but a bit of annoyance when you are the one doing the paying.

**Measurement** - When it comes to gauge a scale and working with very large or small values logarithms come in very hand. This lends itself to many different forms of scales. The Richter scale which is a method for measuring the magnitude of an earthquakes is based on this. A 2 on the Richter scale is 10 times greater than a 1 and 10 times less than a 3. The intensity of sound is measured by the decibel scale which follows this same principle. Another scale called the pH scale follows this method as well and is used to determine the basicity and acidity of a substances.

As we said before logarithms serve the purpose of working with very large or very small values. In the microscopic world that means very large numbers. These values are used to track the growth or decay of bacteria and viruses to help us understand if a medical treatment will work. They are also used to track minuet traces of substances in our drinking water.

**Population Statistics** - We have seen the human population on planet Earth grow by a measure of 1.1% since we have begun to track it. Being able to chart this growth helps us understand how to better prepare for our future in terms of infrastructure and general necessities that we need to ponder. Guess what form of math models all of this?