# Logarithm

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These relations made such operations on two numbers much faster and the proper use of logarithms was an essential skill before multiplying calculators became available.

The equation is fundamental (it implies effectively the other three relations in a field) because it describes an isomorphism between the additive group and the multiplicative group of the field.

To multiply two numbers, one found the logarithms of both numbers on a table of common logarithms, added them, and then looked up the result in the table to find the product. This is faster than multiplying them by hand, provided that more than two decimal figures are needed in the result. The table needed to get an accuracy of seven decimals could be fit in a big book, and the table for nine decimals occupied a few shelves.

The discovery of logarithms just before Newton's era had an impact in the scientific world which can be compared with the invention of the computer in the twentieth century, because many calculations which were too laborious became feasible.

When the chronometer was invented in the eighteenth century, logarithms allowed all calculations needed for astronomical navigation to be reduced to just additions, speeding the process by one or two orders of magnitude. A table of logarithms with five decimals, plus logarithms of trigonometric functions, was enough for most astronomical navigation calculations, and those tables fit in a small book.

To compute powers or roots of a number, the common logarithm of that number was looked up and multiplied or divided by the radix. Interpolation could be used for still higher precision. Slide rules used logarithms to perform the same operations more rapidly, but with much less precision than using tables. Other tools for performing multiplications before the invention of the calculator include Napier's bones and mechanical calculators: see history of computing hardware.

### Calculus

The derivative of the natural logarithm function is

(A proof is shown below.)

By applying the change-of-base rule, the derivative for other bases is

The antiderivative of the logarithm is

See also: table of limits of logarithmic functions, list of integrals of logarithmic functions.

#### Proof of the derivative

The derivative of the natural logarithm function is easily found via the inverse function rule. Since the inverse of the logarithm function is the exponential function, we have . Since the derivative of the exponential function is itself, the right side of the equation simplifies to , the exponential canceling out the logarithm.

## Computers

When considering computers, the usual case is that the argument and result of the function is some form of floating point data type. Note that most computer languages uses for this function while the is typically denoted log10(x).

As the argument is floating point, it can be useful to consider the following:

A floating point value x is represented by a mantissa m and exponent n to form

Therefore

Thus, instead of computing we compute for some m such that . Having in this range means that the value is always in the range . Some machines uses the mantissa in the range and in that case the value for u will be in the range In either case, the series is even easier to compute.

## Generalizations

The ordinary logarithm of positive reals generalizes to negative and complex arguments, though it is a multivalued function that needs a branch cut terminating at the branch point at 0 to make an ordinary function or principal branch. The logarithm (to base e) of a complex number z is the complex number ln(|z|) + i arg(z), where |z| is the modulus of z, arg(z) is the argument, and i is the imaginary unit.

The discrete logarithm is a related notion in the theory of finite groups. It involves solving the equation bn = x, where b and x are elements of the group, and n is an integer specifying a power in the group operation. For some finite groups, it is believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications in public key cryptography.

The logarithm of a matrix is the inverse of the matrix exponential.

A double logarithm, , is the inverse function of the double exponential function. A super-logarithm or hyper-logarithm is the inverse function of the super-exponential function. The super-logarithm of x grows even more slowly than the double logarithm for large x.

For each positive b not equal to 1, the function logb (x) is an isomorphism from the group of positive real numbers under multiplication to the group of (all) real numbers under addition. They are the only such isomorphisms that are continuous. The logarithm function can be extended to a Haar measure in the topological group of positive real numbers under multiplication.

## Notes

1. ↑ James Mills Peirce, The Elements of Logarithms with an Explanation of the Three and Four Place Tables of Logarithmic and Trigonometric Functions (1873).
2. 2.0 2.1 Math Forum, Logarithms: History and Use Retrieved November 20, 2018.
3. ↑ Great Britain Institute of Actuaries, Journal of the Institute of Actuaries and Assurance Magazine, 1873, Vol. 17 (Forgotten Books, 2018, 978-0366971244).
4. ↑ Charles Knight, English Cyclopaedia, Biography, Vol. IV., article "Prony."
5. ↑ MathWorld, Common Logarithm. Retrieved November 20, 2018.

## References

• Great Britain Institute of Actuaries, Journal of the Institute of Actuaries and Assurance Magazine, 1873, Vol. 17. Forgotten Books, 2018. 978-0366971244
• Knight, Charles. The English Cyclopaedia, Vol. 4. Forgotten Books, 2012.
• Peirce, James Mills. The Elements of Logarithms with an Explanation of the Three and Four Place Tables of Logarithmic and Trigonometric Functions. Andesite Press, 2015. ISBN 978-1297495465
• Przeworska-Rolewicz, D. Logarithms and Antilogarithms: An Algebraic Analysis Approach with an appendix by Zbigniew Binderman (Mathematics and Its Applications). New York, NY: Springer, 1998. ISBN 0792349741.
• REA. Math Made Nice & Easy #2: Percentages, Exponents, Radicals, Logarithms and Algebra Basics (Math Made Nice & Easy). Piscataway, NJ: Research & Education Association, 1999. ISBN 0878912010.
• Ryffel, Henry, Robert Green, Holbrook Horton, and Edward Messal. Mathematics at Work. New York, NY: Industrial Press, Inc., 1999. ISBN 0831130830.

All links retrieved November 20, 2018.

• Explaining Math Logarithms.
• Logarithm on MathWorld.
• Jost Burgi, Swiss Inventor of Logarithms.
• Logarithm calculators and word problems with work shown, for school students.
• Logarithms - from The Little Handbook of Statistical Practice.
• Logarithmic Functions

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