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The logarithmic expression is read the logarithm of with base Here, the base is clearly written in the expression. There are two cases in which the base does not need to be written, which will be discussed in this lesson.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Mathematical Operations

The root of a real number expresses another real number that, when multiplied by itself times, will result in Aside from the radical symbol, the notation is made up of the radicand and the index
nth root
However, in the case of a square root, the index can be omitted. Therefore, if in the root there is no index written, it is implied that the index is
Something similar happens with the exponent of a number. An exponent is a number or expression written above a number, indicating that it is being multiplied by itself. The notation is made up of a base — the number being multiplied — and an exponent, which gives the number of times the base appears in the multiplication.
Potenser12 en.svg
The resulting number is commonly called a power. In this example, the base is multiplied by itself times.
However, if the exponent of a number or expression is it does not need to be written.
In the case of logarithms, there are two bases that do not need to be written. Take a look at a calculator and try to identify them. Calculator.jpg
Discussion

Common Logarithm

A common logarithm is a logarithm of base For example, is called the common logarithm of It is equal to because is

the Connection between the base and the exponent for common logarithms

Since common logarithms are used so often, the base does not need to be written.

In the identity above, is a positive number. Recalling the definition of a logarithm, the common logarithm of can be defined for positive values of

Common logarithms can be evaluated using a calculator. For example, to evaluate push enter and then hit

The common logarithm of is about

Pop Quiz

Evaluating Common Logarithms

Use a calculator to evaluate the common logarithms. Round the answer to two decimal places.

evaluate the common logarithm
Example

Earthquakes and Logarithms

A well-known logarithmic scale that is used to express the amount of energy released by an earthquake is the Richter scale.
earthquake
On this scale, each integer number increase indicates an intensity ten times stronger than the previous whole number. For example, an earthquake of magnitude on the Richter scale is times stronger than an earthquake of magnitude An earthquake of magnitude is or times stronger than an earthquake of magnitude The intensity of an earthquake is determined by a seismograph.

Seismograph.jpeg

If two earthquakes have intensity levels and and magnitudes on the Richter scale and the following formula holds true.
In the city of San Francisco was jolted by an earthquake of magnitude on the Richter scale. In nearly years prior, the city had suffered a more powerful earthquake that is estimated to have measured magnitude on the same scale. Vincenzo wants to know how many times more intense the earthquake was than to the earthquake.

Hint

Substitute and for and respectively, in the given formula. Then, use the definition of a common logarithm.

Solution

To calculate how many times more intense the earthquake was compared to the one in the value of needs to be found. To do so, and can be substituted for and respectively, in the given formula.
This equation can be rewritten by following the definition of a common logarithm.
Finally, use a calculator to evaluate
The earthquake was about times more intense than the earthquake.
Example

Using the Properties of Common Logarithms

After studying the relationship between earthquakes and logarithms, Vincenzo became more interested in this fascinating math topic.

San francisco earthquake.jpeg

Now he wants to pair the logarithmic expressions that involve common logarithms with their corresponding simplified expression or number. Help him do this!

Hint

Use the properties of logarithms and the definition of a common logarithm.

Solution

First, simplify the expressions on the left. They can then be paired with their corresponding expressions on the right.

Expression

To simplify this expression, the Quotient Property of Logarithms and the definition of a common logarithm will be used.

The expression should be paired with

Expression

To simplify this expression, the Power Property of Logarithms, the Product Property of Logarithms, and the definition of a common logarithm will be used.

The expression should be paired with

Expression

To simplify the third expression, the Product Property of Logarithms, the Quotient Property of Logarithms, a logarithm identity, and the definition of a common logarithm will be used.

The expression should be paired with

Expression

To simplify the second to last expression, the Quotient Property of Logarithms, the Power Property of Logarithms, and the definition of a common logarithm will be used.

The expression should be paired with

Expression

Finally, to simplify the last expression, the Product Property of Logarithms and the Power Property of Logarithms will be used.

The expression should be paired with
Explore

Calculating the Value of an Expression for Bigger Values of

Consider the following algebraic expression.
This expression can be evaluated for different values of Pay close attention to the value of the expression as increases.
(1+1/n)^n for bigger values of n
What conclusion can be made?
Discussion

The Natural Base

Logarithms and exponents are closely related. This part of the lesson will explore a particular type of exponential expression. Recall the formula for compound interest.
Compound interest formula
Here, is the interest rate in decimal form. If the interest rate is which is extremely profitable, the value of equals
The number is the amount of times the interest is compounded each year. This is how often the accrued interest is added to the balance. The more often the interest is compounded, the higher the profit will be each year. What happens as increases? To examine this, the formula will be rewritten using the the Power of a Power Property.
In the revised formula, the expression in the outer parentheses is a constant that depends only on To analyze what happens as  increases, larger and larger values will be substituted into this expression.
(1+1/n)^n for bigger values of n
As the values of increase higher and higher, the expression seems to approach a value of about In fact, as goes to infinity, the value of the expression approaches a constant that is called the natural base, or
Concept

The Number

The number — commonly called the natural base — is an irrational mathematical constant named by the mathematician Leonhard Euler.

Euler's number appears in several areas of mathematics and has multiple uses. For example, it is often used as the base of exponential and logarithmic functions.
Discussion

The Natural Logarithm

A natural logarithm is a logarithm with base . Although it is correct to write the natural logarithm is more commonly written as

This means that equals the exponent to which must be raised to equal

Natural logarithms can be evaluated using a calculator. For example, to evaluate push input and then hit

The natural logarithm of is about
Pop Quiz

Evaluating Natural Logarithms

Use a calculator to evaluate the natural logarithms. Round the answer to two decimal places.

evaluate the natural logarithm
Example

Using the Properties of Natural Logarithms

Vincenzo already knows how to use the properties of logarithms when dealing with common logarithms. Now he wants to use the properties when natural logarithms are involved.
Properties for common and natural logarithms
To do so, Vincenzo asked his math teacher for some extra practice. She asked him to identify the only expression that is not equivalent to the others. Help him do this!

Hint

Simplify all the expressions using the properties of logarithms.

Solution

Simplify the expressions so that the expression that is not equivalent with the others can be easily identified.

Expression

To simplify this expression, the Product Property of Logarithms, the Quotient Property of Logarithms, and the Power Property of Logarithms will be used.

The expression is equivalent to

Expression

To simplify the second expression, the Product Property of Logarithms, the Power Property of Logarithms, and the definition of a natural logarithm will be used.

The expression is also equivalent to

Expression

To simplify the third expression, the Quotient Property of Logarithms and the definition of a natural logarithm will be used.

The expression is also equivalent to

Expression

To simplify the second to last expression, the Product Property of Logarithms and the Quotient Property of Logarithms will be used.

The expression is equivalent to Therefore, this expression is not equivalent to the previous expressions.

Expression

The last expression will be simplified by using the Power Property of Logarithms and the Product Property of Logarithms.

The last expression, is also equivalent to Therefore, the expression that is not equivalent to the others is the fourth expression.
Discussion

Change of Base Formula

Most calculators only calculate common and natural logarithms. These are logarithms with base or Luckily, there is a formula that allows any logarithm to be written in terms of common or natural logarithms.

Rule

Change of Base Formula

A logarithm of arbitrary base can be rewritten as the quotient of two logarithms with the same base by using the change of base formula.

This rule is valid for positive values of and where and are different than

Proof

Let Therefore, by the definition of a logarithm, it is known that
By the Reflexive Property of Equality, is equal to itself.

Note that this formula is helpful to calculate any logarithm using a calculator, since the new base can be any positive number different than This means that the new base can be or

Example

Using the Change of Base Formula

Vincenzo has discovered that he can use the Change of Base Formula to calculate the value of any logarithm.
Vincenzo at the computer with a thought bubble where the Change of Base Formula is written
Help him answer the following questions!
a Use common logarithms and a calculator to find the value of Round the answer to two decimal places.
b Use natural logarithms and a calculator to find the value of Round the answer to two decimal places.

Hint

a By the Change of Base Formula,
b By the Change of Base Formula,

Solution

a Recall the Change of Base Formula.
Here, and are positive numbers, with and different than While the base can be any number, in this case it might be most helpful if it was Then the logarithm of with base could be expressed in terms of common logarithms.
The value of the numeric expression on the right-hand side of the above equation can be found by using a calculator.

Therefore, the value of rounded to the nearest hundredth is

b Once again, recall the Change of Base Formula, this time with natural logarithms.
Here, and are positive numbers, with and different than For this exercise, let be Then the logarithm of with base can be expressed in terms of natural logarithms.
The value of the numeric expression on the right-hand side of the above equation can be found by using a calculator.

Therefore, the value of rounded to the nearest hundredth is

Closure

Leonhard Euler and His Contributions

In this lesson, the Swiss mathematician Leonhard Euler was mentioned. Euler was also a physicist, astronomer, geographer, logician, and engineer. During his life, Euler came up with principles that set the foundations for most of the mathematics used nowadays. He was a revolutionary thinker in the fields of geometry, calculus, trigonometry, differential equations, and number theory.

Euler.jpeg

Among other important contributions, he introduced most of the mathematical notations that are used today. Euler was the first person to use the letter to denote the base of a natural logarithm. Furthermore, although he was not the first to use it, Euler popularized the use of the Greek letter to indicate the ratio of the circumference of a circle to its diameter.