What is a logarithm?

What is a logarithm? In mathematics, the logarithm to the base b of a positive number y is defined as follows: If y = b^x, then log_b y = x

Read log_b y as " log base b of y "

Just like we saw in the lesson about exponential function, b is not equal to 1 and b is bigger than zero.

The exponent x in the exponential expression b^x is the logarithm in the equation log_b y = x

What is a logarithm in easy terms?

For example, what is the logarithm of the expression log₅ 25?

2 is the logarithm of the expression log₅ 25. Why? In the expression, we can see that the base is 5.

Therefore, ask yourself, "5 to the power of what is equal to 25?"

Since 5² = 25, log₅ 25 = 2.

What is the logarithm of the expression log₂ 16 ?

Since 2 to the fourth power is equal to 16, log₂ 16 = 4.

Types of logarithms

The two types of logarithms are the common logarithm and the natural logarithm.

What is a common logarithm? 

A common logarithm is a logarithm that uses base 10. Therefore, the expression log_b y = x becomes log₁₀ y = x

As a result, you are always looking for the number of times you multiply 10 by itself to get y.

You can write the common logarithm as log₁₀ y or as log y

What is log₁₀ 1000?

Since 10 to the third power = 1000,  log₁₀ 1000 = 3.

What is a natural logarithm? 

A natural logarithm, also called natural log is a logarithm that uses base e where e = 2.718281828.

Therefore, the expression log_b y = x becomes log_e y = x

As a result, you are always looking for the number of times you multiply e by itself to get y.

You can write the natural logarithm as log_e y or as ln y. However mathematicians have agreed that the natural logarithm can have its own notation. In most cases we write the natural logarithm as ln y without showing the base e although we know it is there.

What is ln 10?

You are looking for a number x such as (2.718281828)^x = 10

You should use a calculator since it is not so easy to find. This lesson shows how to find the natural log using a calculator.

Writing in logarithm form using the exponential form

Write 49 = 7² in logarithm form

Write the definition

If y = b^x, then log_b y = x

Substitute

If 49 = 7², then log₇ 49 = 2

The logarithmic form of 49 = 7² is log₇ 49 = 2

Write 1/8 = (1/2)³ in logarithm form

Write the definition

If y = b^x, then log_b y = x

Substitute

If 1/8 = (1/2)³, then log_1/2  1/8 = 3

The logarithmic form of 1/8 = (1/2)³ is log_1/2  1/8 = 3

What is a logarithm used for?

Among others, scientists use logarithms to perform the following tasks:

• Measure acidity in substances called pH
• Find the magnitude of earthquakes
• Find interest rate

Logarithm, earthquake, and the Richter scale

For example, an earthquake that has a magnitude of 7.9 is 11600 times as strong as an earthquake that occurred in the Caribbean. Find the magnitude of the earthquake that occurred in the Caribbean.

Let x be the magnitude of the earthquake that occurred in the Caribbean.

Then, 10^7.9 / 10^x = 11600

10^{7.9 - x} = 11600

log₁₀ 11600 = 7.9 - x

4.0644579892 = 7.9 - x

x = 7.9 - 4.0644579892

x = 3.83554201

What is a logarithm? FAQs

What is the logarithm of a number?

The logarithm of a number is simply an exponent or the number of times a base is multiplied by itself to get the number. For example, log₆ 36 = 2. In other words, the base 6 must be multiplied by itself twice(exponent is 2) to get 36.

What is a logarithm with base 10 called?

The logarithm with base 10 is called the common logarithm or the decimal logarithm. For example, log₁₀ 1000 = 3 is a common logarithm.

Can a logarithm be negative?

Yes, a logarithm can be negative. For example, log₄ 0.25 = -1 In other words, 4^-1 = 1/4 = 0.25. While the logarithm or exponent can be negative, you cannot take the logarithm of a negative number. For example, log₁₀ -100 = does not exist. Check this lesson to learn more.