Logarithm

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**Logarithms** or **logs** are a part of mathematics. They are related to exponential functions. A logarithm tells what exponent (or power) is needed to make a certain number, so logarithms are one of the inverse operations of exponentiation (the other one being roots). Historically, they were useful in multiplying or dividing large numbers.

An example of a logarithm is \log_2 8=3. In this logarithm, the base is 2, the power is 8 and the answer is 3. In this case, the exponentiation function would be:

2^3=2\times2\times2=8

The most common types of logarithms are **common logarithms**, where the base is 10, **binary logarithms**, where the base is 2, and **natural logarithms**, where the base is e&nbsp;\approx2.71828.

## History
Logarithms were first used in India in the 2nd century BCE. The first to use logarithms in modern times was the German mathematician Michael Stifel (around 14871567). In 1544, he wrote down the following equations: q^m q^n=q^{m+n} and \frac{q^m}{q^n}=q^{m-n}. This is the basis for understanding logarithms. For Stifel, m and n had to be whole numbers. John Napier (1550–1617) did not want this restriction, and wanted a range for the exponents.

According to Napier, logarithms express ratios: a has the same ratio to b, as c to d if the difference of their logarithms matches. Mathematically: \log a-\log b=\log c-\log d. At first, base e was used (even though the number had not been named yet). Henry Briggs proposed to use 10 as a base for logarithms, as such logarithms are very useful in astronomy. Some authors prefer the use of natural logarithms as \log x, (and usually mention it on preface pages). There is no single standard of what "\log x" means, so it is important to see how each author defines it.

## Common bases for logarithms 
## Properties of logarithms 
Logarithms have many properties. For example:

### Properties from the definition of a logarithm
This property is straight from the definition of a logarithm:
:\log_n(n^a)=a

For example
:\log_2(2^3)=3, and
:\log_2(\frac12)=-1, because \frac12=2^{-1}.

The logarithm to base b of a number a, is the same as the logarithm of a divided by the logarithm of b. That is,
:\log_b a=\frac{\log a}{\log b}
For example, let a be 6 and b be 2. With calculators we can show that this is true (or at least very close):
:\log_2 6=\frac{\log 6}{\log 2}
:\log_2 6\approx2.584962
:2.584962\approx\frac{0.778151}{0.301029}\approx2.584970
The results above had a small error, but this was due to the rounding of numbers.

Since it is hard to picture the natural logarithm, we find that, in terms of a base ten logarithm:
:\ln x=\frac{\log x}{\log e}\approx\frac{\log x}{0.434294}, where 0.434294 is an approximation for the logarithm of e.
### Operations within logarithm powers
Logarithms which multiply inside their power can be changed as follows:
:\log ab=\log a+\log b
For example,
:\log1000=\log(10\cdot10\cdot10)=\log10+\log10+\log10=1+1+1=3
Similarly, logarithm which divides inside the argument can be turned into a difference of logarithm (because it is the *inverse operation* of multiplication):
:\log(\frac ab)=\log a-\log b
### Logarithm tables, slide rules, and historical applications
Before electronic computers, logarithms were used every day by scientists. Logarithms helped scientists and engineers in many fields such as astronomy.

Before computers, the *table of logarithms* was an important tool. In 1617, Henry Briggs printed the first logarithm table. This was soon after Napier's basic invention. Later, people made tables with better scope and precision. These tables listed the values of log_b x and b^x for any number x in a certain range, at a certain precision, for a certain base b (usually b=10). For example, Briggs' first table contained the common logarithms of all integers in the range 1–1000, with a precision of 8 digits. 

Since the function f(x)=b^x is the inverse function of log_b x, it has been called the antilogarithm. People used these tables to multiply and divide numbers. For example, a user looked up the logarithm in the table for each of two positive numbers. Adding the numbers from the table would give the logarithm of the product. The antilogarithm feature of the table would then find the product based on its logarithm.

For manual calculations that need precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier ways.

Many logarithm tables give logarithms by separately providing the characteristic and mantissa of x, that is to say, the integer part and the fractional part of log_{10}x. The characteristic of 10\cdot x is one plus the characteristic of x, and their significands are the same. This extends the scope of logarithm tables: given a table listing log_10 x for all integers x ranging from 1 to 1000, the logarithm of 3542 is approximated by
:\log_{10}3542=\log_{10}(10\cdot354.2)=1+\log_{10}354.2\approx1+\log_{10}354

Another critical application was the slide rule, a pair of logarithmically divided scales used for calculation, as illustrated here:

Numbers are marked on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms. For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. Many engineers and scientists used slide rules until the 1970s. Scientists can work faster using a slide rule than using a logarithm table.

## Related pages 
* Euler's number
* Prime number theorem

## References