# Express as a Single Logarithm Worksheets

There are plenty of times where we will need to simplify or reduce equations or expressions that we are working with in order to take our next step forward towards solving the problem or system. On of the most common ways to approach this is by finding something in common that sticks out to you. This can be in the form of a factor, constant, or variable. In this topic we show you how to apply this to logarithms. We will also introduce a few new properties that you may not be familiar with. These worksheets and lessons help students understand how to convert logarithm that are undergoing operations into a single value.

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

- Sums and Difference of Logs Step-by-Step Lesson- See if your students can come up with a good explanation for the image.
- Guided Lesson - Just a bunch more logs to throw on the fire. By fire, I mean your brain.
- Guided Lesson Explanation - The process is very similar regardless of the operation that is included in the mix.
- Practice Worksheet - Combining the operations found in an algorithm is just so interesting.
- Matching Worksheet - Match the values to the log operations that have created them.
- Express as a Single Logarithm Five Pack of Worksheets - It is kind of like saying, "Combine Like Terms!" Almost!

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

### Homework Sheets

We take on this topic from a slightly different angle than the lessons and previous practice worksheets.

- Homework 1 - You will run through this equation and create a solution.
- Homework 2 - List all of the possible solutions which may include a positive and a negative value.
- Homework 3 - We introduce logarithmic products in with these equations.

### Practice Worksheets

You will mostly see logarithmic sums in this set of equations, but we make room for differences as well.

- Practice 1 - You may not need the log table to solve all of these.
- Practice 2 - Note where the constants are placed in some of these problems. They will help you better understand what is being asked of you.
- Practice 3 - Remember back to what you learned about the property of logarithms.

### Skill Quizzes

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

- Quiz 1 - I find that students always forget to place the symbol (±) for cases where two answers may exist.
- Quiz 2 - This is a quiz that I would use to see if they know what they are doing.
- Quiz 3 - Some of the problems on this quiz are a bit more difficult.

### How to Express Mixed Log Values as a Single Logarithm

A logarithm (or simply 'log') is a mathematical tool that is used to solve complex exponential equations using a relatively simpler technique. It can be applied in long expressions like x^{3} - y^{2} + z^{5}. There is a certain method of expressing exponential expressions in terms of logs. First, the exponent is written, then log and then the variable part. For example, in this case, the equation will become 3log(x) - 2log(y) + 5log(z) after it is written with log. But how to express these long expressions as a single logarithm?

There are a few logarithmic identities. When two log terms are being subtracted, we can divide them by each other and write them with log as a single log term. Similarly, if two log terms are being added to each other, we can multiply them with each other and write them with log as a single log term. In the case discussed above, after the implementation of log identities, the expression will look like this:

x^{3} - y^{2} + z^{5} = 3log(x) - 2log(y) + 5log(z) = log(x^{3}) - log(y^{2}) + log(z^{5}) = log(x^{}3/y^{2}) + log(z^{5}) = log((x^{3}/y^{2}) . z^{5})

Here is how mixed log values are expressed as a single logarithm. There are a number of different logarithm rules that you can apply to these situations to help you shore up everything to express mixed values as a single logarithm. Here are the most commonly used rules:

**Power Rule** - This comes in helpful when you are working with logarithms of values that contain an exponent. When you are working on logarithm of an exponential number you need to realize that it is the equivalent of the exponent multiplied by the logarithm of the base. Mathematically we can state this as:

log_{b} (M^{k}) = k × log_{b} M

**Product Rule** - This states that the product of numbers is the same value as sum of logarithms of the individual numbers. It can be stated as this equation:

log_{b} (M × N) = log_{b} M + log_{b} N

**Quotient Rule** - Just as quotient operations are the inverse of multiplication operations, so is this rule to the previous one we learned. The logarithm of a quotient of numbers is the same as the difference of the logarithm of those same numbers. It can be stated as this equation:

log_{b} (M ÷ N) = log_{b} M - log_{b} N

**Zero Rule** - The last rule is one that you rarely come across, but when you do it saves the day. It is a situational rule that involves finding the logarithm of 1. It simply states that if you are finding the logarithm of 1 and b is greater than 0, but not equal to 1 the overall value is 0. We can state it as:

log_{b} (1) = 0