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
Keep in mind that whenever you are looking for the logarithm, you are looking for an exponent, or the number that tells how many times the base is multiplied.
For example, what is the logarithm of the expression log_{5} 25?
2 is the logarithm of the expression log_{5} 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 5^{2} = 25, log_{5} 25 = 2.
What is the logarithm of the expression log_{2} 16 ?
Since 2 to the fourth power is equal to 16, log_{2} 16 = 4.
The two types of logarithms are the common logarithm and the natural logarithm.
A common logarithm is a logarithm that uses base 10. Therefore, the expression log_{b} y = x becomes log_{10} 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 log_{10} y or as log y
What is log_{10} 1000?
Since 10 to the third power = 1000, log_{10} 1000 = 3.
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.
Write 49 = 7^{2} in logarithm form
Write the definition
If y = b^{x}, then log_{b} y = x
Substitute
If 49 = 7^{2}, then log_{7} 49 = 2
The logarithmic form of 49 = 7^{2} is log_{7} 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
Among others, scientists use logarithms to perform the following tasks:
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.
The Richter scale is commonly used to measure the magnitude of earthquakes. In the Richter scale, the common logarithm is used and adding a magnitude of 1 makes the earthquake 10 times more powerful. An earthquake of magnitude 7 is 1000 times more power an earthquake of magnitude 4 since 10^{7} / 10^{4} = 10^{7-4} = 10^{3 }= 1000
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
log_{10} 11600 = 7.9 - x
4.0644579892 = 7.9 - x
x = 7.9 - 4.0644579892
x = 3.83554201