Log (Logarithm) calculator
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Log Calculator
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What is a Log(Logarithm)?
Logarithm (or log) is a mathematical operation which is the inverse of exponentiation. This means that, the logarithm of a number is the power that the base must be raised to, to obtain this number. For example, if ‘x’ is the number obtained when the base ‘b’ is raised to the power ‘y’, then the log operation will yield the number ‘y’, which is the power that the base ‘b’ must be raised to, in order to obtain ‘x’.
The exponentiation is written as x = b^{y}. The corresponding log notation is y = log_{b} x. This is read as ‘the logarithm of x to the base b’. This basically shows that the number base ‘b’ must be raised to the power ‘y’ to obtain the value ‘x’. For example, 8 = 2^{3}, so 3 = log_{2} 8. The logarithm can be calculated for only positive real numbers, and the base is a positive number not equal to 1.
The log notation is a very convenient way of expressing large numbers.
There are two types of logarithms: the common logarithm and natural logarithm. In usual notations, the word ‘log’ denotes the common logarithm where the base is 10, although any base can be used. For the common logarithm, if the base 10 is used, it is sometimes not explicitly mentioned, and is denoted only as log. For example, log_{10} 5 is simply written as log 5. For any other number, the base is always mentioned.
The other kind of logarithm is natural log, where the base is e, which is Euler’s number. This is equal to 2.7182. In this case, the logarithm is denoted by ln or log with e as the base. For example, log_{e} 5 = ln 5. The natural logarithm defines how many times e needs to be multiplied with itself to get 5.
The idea of logarithms was proposed in the 17th century by Scottish mathematician, John Napier for ease of computations.
Properties of logarithm
Given below are some important properties of logarithm.
Base – Denoted by a subscript, the base of a logarithm ‘b’ is the number which needs to be multiplied with itself a certain number of times to get the number ‘x’. When the base is Euler’s number, the logarithm is called natural logarithm.
Logarithm – The logarithm of a number is the power that the base must be raised to, to obtain this number. If ‘x’ is the number obtained when the base ‘b’ is raised to the power ‘y’, then the log gives ‘y’, which is the power that the base ‘b’ must be raised to, in order to obtain ‘x’.
Logarithm formula
The formula for logarithm is y = log_{b} x, which is equivalent to x = b^{y} in exponent form.
Usually, log alone indicates that the base 10 is being used. When the base e is used, then log_{e} or ln is used. log_{2} is known as the binary logarithm, is used in computer systems. The logarithm can be calculated for only positive real numbers, and the base must be a positive number not equal to 1.
The logarithm of 0 is not defined. The logarithm of 1 is 0 for any base, that is log_{b} 1 = 0. The logarithm of the base is 1, that is, log_{b} b = 1.
The next section lists the various rules followed by logarithms.
Logarithm rules
Logarithms follow several rules which have been listed below.
Rule | Example |
---|---|
Product rule: $$ log_b (x * y) = log_b \,x + log_b \,y $$ | For example: $$ log_2 (4 \,*\, 8) = log_2 \,4 + log_2 \,8 = 2+3 = 5 $$ |
Quotient rule: $$ log_b ({x \over y}) = log_b \,x - log_b \,y $$ | For example: $$ log_2 ({4 \over 8}) = log_2 \,4 - log_2 \,8 = 2-3 = -1 $$ |
Power rule: $$ log_b (x^y) = y \; *\;log_b \,x $$ | For example: $$ log_3 (5^2) = 2\;*\;log_3 \,5 $$ |
Switching the bases of a logarithm: $$ log_b \;x\; = {1\over {log_x b}} $$ | For example: $$ log_2 \;3\; = {1 \over {log_3 2}} $$ |
Changing the base of a logarithm: $$ log_b \;x\; = {log_c \;x \over log_c \;b} $$ | For example: $$ log_2 \;3\; = {log_10 \;3 \over log_10 \;2} $$ |
The logarithm of a negative number is undefined. | |
The logarithm of 0 is undefined. | |
The logarithm of 1 is 0 for any base, that is log_{b} 1 = 0. For example, log_{3} 1 = 0. | |
The logarithm of the base is 1, that is, log_{b} b = 1. For example, log_{3} 3 = 1. | |
For the natural logarithm, log_{e}(e^{e}) = ln e^{x} = x. |
Areas of application
Logarithms are frequently used when dealing with large numbers. Some examples are:
1) Log can be used to model growth, like when a sum of money deposited in a bank increases over time. It shows the effect of growing and the cause can be found using logarithms. If $500 increase to $600 over a period of 5 years, log helps us calculate the rate of continuous return as: $$ rate = {{ln({600\over500})}\over5} = 0.036 = 3.6% $$ Hence, an interest rate of 3.6% is responsible for this change. So, logarithms can be used to compute interest rates.
2) Logarithms with base 2 are used in computers. This is known as the binary system.
3) Large numbers, like millions and billions, can be expressed very conveniently using log with the base 10. For example, 1 million = 10^{6} can be written as log_{10} 1000000 = log_{10}(10^{6}) = 6 and 1 billion can be written as log_{10} 1000000000 = log_{10}(10^{9}) = 9.
4) They are used to model situations involving exponential growth and decay.
5) They are used in the Richter scale and Decibel. The Richter scale measures the intensity of earthquakes on a single scale with a small range (usually from 1 to 10). The increase of each point shows that the intensity of the earthquake is increasing. Decibels are also used in the same way, but they can be negative. The range of sound can go from extremely quiet (like a pin drop) to extremely loud (like a supersonic aircraft).