### Logarithm Equations Worksheets By Specific Topic Area

We trimmed these down to focus on skills that are found in the Core curriculum. These are all new and improved to match the standards. We will be adding many more all the time. You will find these super helpful. Below you will basic to advanced worksheets.### Basic Logarithm Worksheets

- Logarithmic Format - This is a really good starter for kids. You can help your students transition to the concept of logs pretty quickly with this one.
- Logarithm Operations Worksheet - Using logs in operations form is usually difficult for many students please allow extra time for them to review this skill.
- Expanding Logarithms - This is great to use as a refresher or Do Now activity. The work space helps focus the students.
- Expand the Logarithm Using the Quotient Rule - This another skill that recently came to light again. Just follow the quotient rule formula and you should be good to go.
- Express in Exponential Form - Have students rewrite each express. Make sure that they double check number 4, it's a bit tricky for some.
- Finding Log Values (w/Calculator) - Another quick calculator workout for you! I would practice this a few times.
- The Value of the Argument in Logarithms - Students are asked to find the outcome of the argument each time. This is a moderately difficult task for students.
- Using a Calculator to Find Logarithm Values - Here is some practice for your calculator and logs. Help them locate the proper buttons. These values can be cryptic on several versions of calculators.
- Using the Change of Base Rule - You might find this skill very deceiving. It seems extremely easy, but it does get very difficult at times.

### Logarithm Operation Worksheets

- Advanced Logarithm Operations - This can be a very difficult skill for many children. Always make sure to check for understanding often with this set.
- Antilogarithms - We have to be honest, our math staff hadn't seen this skill in over 20 years, until it popped back up in the math core curriculum recently.
- Combining Logarithms - Perform a number of different operations on a basic log scale. Got to love the surfing puppy!
- Tricky Exponents: Solving For X - These can take a good amount of time each. Students usually take 2 - 3 minutes a problem.

### Logarithm Word Problems

Please note that you will need to remind students to solve these problems using your preferred format. These sheets have many different applications.

- Logarithms Word Problems - These problems can be solved using a wide variety of means, but we would like to remind students to focus on using Log solutions.
- Decay and Practical Everyday Logarithms - These types of problems are very often found in the Chemistry classroom.
- Mixed Logarithms Word Problems - This is a great variety of Log based questions.
- An Introduction to Logarithms Word Problems - Four quick and baseline questions for students new to logs in a story setting. It is a great starter!

### Natural Logarithm Worksheets

- Natural Logarithm - Using Calculators - Not all calculators are created equal when it comes to this set of worksheets. It's like they remove the ln button in Ancient Roman times; even though it was about 5 years ago I remember last seeing them on standard calculators.
- Natural Logarithm Equations and Finding Unknowns - Be careful these can be very tricky at first. Make sure you master that first problem and everything should come with it.
- Solving Natural Logarithms Equations - This is about as tough as the CORE gets with Logarithms. Try to minimize your operations to make it easier for yourself.
- Natural Logarithm Operations - You can have them solve number if they wish. The answer key that we provide work it down to natural log of a fraction, in most cases.

## All About Logarithms

A logarithm is a math technique that defines the number of times a specific number, known as the base, is multiplied by itself to obtain another number. Since logarithms connect geometrical and mathematical progressions, samples of them may be derived from nature, and artwork, such as guitar fret spacing, material hardness, and the intensity of noises, lights, thunderstorms, quakes, and chemicals. Logarithms even explain how individuals learn about numbers naturally.

Let's take a look at the concept of an logarithm, the different types, rules, and properties that are related to the concept.

**What is a Logarithm?**

A logarithm is just another term to describe exponents. It can be used to solve equations that can't be addressed using only exponents. The concept of logarithm is not complicated to understand. To comprehend a logarithm, remember that a logarithmic expression is simply another way of stating an exponential expression. Exponent and logarithm are the inverses of one another.

For instance, we know 3 raised to the power 2 is 9. If you convert this statement into an exponential expression, it will become 3^{2} = 9.

But what if someone asks, "3 raised to the power which number is 9". We know the answer is 2, but how do we express this in a mathematical equation. Here is how:

3^{2} = 9 =>log_{3}(9) = 2

The above mentioned is a logarithmic equation, which can be expressed as "log base three of nine is two."

Both expressions represent the same numerical connection between the numbers 3, 2, and 9, where 2 is the exponent, and 3 is the base.

However, the distinction is that the exponential expression concentrates on the power, 9, while the logarithmic expression focuses on the exponent, 2.

We can use an exponential expression to generate a logarithmic expression.

a^{b} = c =>log_{a}(c) = b

In the above expressions, both have highlighted the numerical connection between a, b and c

However, in the logarithmic expression:

- a is the base

- b is the exponent

- and c is the argument

The expression on the right-hand side of the arrow states, "The log of c to the base a equals b."

Take note that 'a' is the base of expressions on both the left and right sides of the arrow. Notice how the base a and exponent b do not remain on the same end of the equation in a logarithmic expression.

**Types of Logarithms**

In just about all circumstances, we are dealing with two forms of logarithms. The come in natural and common forms.

**Natural**

Another name for the natural log is base e log. A natural logarithm is denoted by the symbols ln or log. The "e" stands for Euler's constant, which is roughly equal to 2.71828. A natural logarithm of 87, for example, is expressed as ln 87. The natural logarithm specifies how many times "e" must be multiplied to obtain the correct result.

Here are a few examples:

ln (87) = 4.4659.

Here the base e logarithm of 87 is equal to 4.4659.

ln (978) = 6.885

Here the base e logarithm of 978 is equal to 6.885.

**Common**

Another name for common logarithms is base ten logarithms. It is denoted by log10 or just log. For instance, the common log of ten thousand is expressed as a log (10000). The common logarithm specifies the number of times 10 must be multiplied to obtain the appropriate result.

Here are a few examples:

log(100) = 2

log (1000) = 3

log (10000) = 4

In the equations above, if we multiply the number 10, two, three, and four times, we get 100,

1000, and 10000, respectively.**Rules**

Let's learn the basic logarithmic rules and how to apply them to restructure logarithmic equations.

**The Product Rule**

The log of a product is the total of the logs of its variables, according to this property.

Logarithmic equations can be rewritten using the product rule.

log_{b}(xy) = log_{b}(x) + log_{b}(y)

It is worth noting that the base of all logs will remain the same throughout the expression. The log product rule is developed from the exponent product rule.

b^{x}⋅b^{y} = b x+y

Let's look at an example of the product rule of the log:

log6(4x) => log6(4) + log6(x)

**The Quotient Rule**

According to this characteristic, the log of a fraction or quotient is the difference between the logs in the expression.

log_{b} (x/y) = log_{b} (x) - log_{b} (y)

It is worth noting that the base of all logs will remain the same throughout the expression. This is similar to/derived from the exponent quotient rule:

b^{x} / b^{y} = b^{x}-y

Let's look at an example of the quotient rule of the log:

log_{4} (3/x) => log_{4} (3) - log_{4} (x)

**The Power Rule**

According to this rule, the logarithm of a power is the exponent multiplied by the log of the power's base.

log_{b}xy = y log_{b} x

The bases should be the same along both sides in this instance. This is similar to the power of power rule of exponential expression:

(bx)^{y} = b^{xy}

Here is an example of the power rule of the log:

log_{3} x 2 => 2 log_{3} (x)

**Log One Rule**

Regardless of the base, the output of log 1 is 0. Since we understand given exponential characteristics that x^{0} = 1 for any 'x'. When converted to logarithmic form, log_{x} 1 = 0 for any 'x'.

Here are a few examples of the log one rule:

log_{20} (1) = 0

log_{4567} (1) = 0

If we apply it to the natural log, we get ln 1 = 0 since e^{0}.

**Wrapping Up**

Logarithms are derived from the properties of exponents. One can easily understand the concept of logarithm if they have thoroughly understood exponents. Logarithms are employed to perform the most complicated multiplication and division operations. They are used mainly by mathematicians and physicists to solve complex equations.