The logarithm (log) of a number refers to the exponent or power through which another value should be raised for producing an equivalent value of the same number. The logarithm theory was introduced by a Scottish mathematician called John Napier in the 17th century. Let’s know more about Logarithm vs Natural Log.
Later, engineers, navigators, and scientists have adopted this concept for performing computation incorporating logarithmic tables. The log of a number can be expressed as follows;
logbN=x
Here, the base is ‘b’ which can be any digit except 0 and 1. N and x are the argument and the exponent, respectively.
This can be further explained easily with an example.
The log of 32 with base 2 is five, which can be shown as;
log232=5
Hence, the base of the logarithmic equation should be a digit beside 0 and 1. Now, let us learn more about the two other logarithms which are used frequently in maths. These are natural and common logarithms.
Natural Logarithm: Meaning and Definition
The natural log of a number (N) is the exponent or power to which ‘e’ needs to be raised to match the value of N. ‘e’ is the constant, known as the Napier constant, which has an approximate value of 2.718281828.
N=ex is going to be the same as N=x.
The natural logarithmic function is used in pure mathematics like calculus.
The properties of logarithms are similar to those of natural logarithms.
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Product Rule
⇒ In(ab) = In (a) + In (b)
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Quotient Rule
⇒ In(a/b) = In (a) – In (b)
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Reciprocal Rule
⇒ In(1/a) = -In (a)
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Power Rule
⇒ In(ab) = b In(a)
Some of the other properties include:
- In(1) = 0
- ln (∞) = ∞
- In(e) = 1
- In (ex) = x
- ein(x) = x
Graphing and scientific calculators have keys for natural and common logarithms. The calculator has a key labelled “In” for common logarithm and “e” for natural log.
Common Logarithm: Meaning and Definition
A common log has 10 as its fixed base. The common log of the number (N) can be written as;
log N or log 10N
Sometimes, common logarithms are also referred to as decimal logarithms and decadic logarithms.
So, if log N=x, then this logarithmic equation can be represented in exponential form, 10x=N.
Henry Briggs, a British mathematician, introduced common logarithms in the 18th century. Hence, they are also known as the Briggsian logarithms. Common logarithms are used in both the engineering and science field.
For instance, the alkalinity and acidity of a substance can be expressed in exponential.
Some of the properties of the common log are similar to that of all logarithms. These are as follows,
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Quotient Rule
The quotient of two common log values will be equal to the difference of each common log.
⇒ log(m/n)= log m – log n
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Product Rule
The sum of common logs individually will be equal to the product of two common logs.
⇒ log(m n) = log m + log n
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Zero Exponent Rule
⇒ log 1= 0
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Power Rule
The product of an exponent and the common log will be equal to the common log of a number.
⇒ log(mn)= n log m
To know more about common and natural logarithms, check out Logarithm vs Natural Log.
Real-life Applications of Logarithms
One of the classic examples of logarithms in real life is the Richter Scale used for measuring earthquakes. Another interesting fact about using the logarithmic scale is that it helps determine the fault line’s length.
The largest earthquake that was ever recorded had a magnitude of 9.5. The earthquake happened in Chile on May 22, 1960, which created a long fault line of 1000 miles.
Besides that, some other common uses of logarithmic scales are pH (for pool water testing), light intensity, and decibels.
Applications of Logarithms in terms of Data Science
There are various uses of logarithms when it comes to the field of data science. Some of them are as follows,
- Logarithmic derivatives are cleaner to use for complicated functions. It is an extremely effective application of logarithms.
- Logarithmic transformations are even extremely useful to determine patterns in a given set of data. As logarithmic transformation helps straighten out a function and turn it into an exponential function, it makes it easier for data scientists to read and understand data more easily.
- Lastly, log odds also play a vital role in terms of logistic regression. This means a probability can be converted easily to log odds by taking the log and determining the odds ratio.
Wrapping Up
Hence, logarithms can be used for modelling different phenomena. It is a handy tool that one can add to their mathematical toolbox.
