Rules for logarithms

Corresponding to the earlier stated features of the exponential function, let in x=log_ga or g^x=a the base g > 0 and 1and always a> 0. Moreover, let us not change the base g, so that we need not specify it when writing the symbol log.

We know and that the equations
and are equivalent.

Rule I: loga + logb. Take the power of g to loga + logb and apply the power rule a^m+n = a^m·aⁿ, to obtain

.

Since the equations and are equivalent, they yield
g^{log a + log b} =ab, loga + logb = log ab :

logab = loga + logb.

The expansion to several factors is

logabc = loga + logb + logc.

Rule II: log a - logb. Take the power of g to log a - log b and apply the power rule a^m-n = a^m/aⁿ:

,

whence

and

.

Rule III: nloga. Form g^nloga and apply the power rule a^m-n = (a^m)ⁿ, to obtain

,

whence follows

.

Rule IV: This rule also applies if n is a fraction. For practical reasons, it is useful to emphasize here the special case when n = 1/m:

.

Examples:

These rules show that by transition to logarithms

In other words, the types of computation are reduced to a lower stage. For example, in order to compute 4.238⁴⁸¹, you had formerly to multiply the base 4.238 with itself 480 times. If you assume that without a modern computer a multiplication takes half a minute, you would have used 4 hours to obtain this result and be subject to many errors underway. With logarithms, the task becomes a single multiplication. When we talked about taking roots, considerations were limited to square roots. For higher order roots, for example , we had to use the formula (a + b)⁷ and complicated manipulations. Now you obtain the result by a single division, whence it will be worthwhile to pursue the trend of thought in detail.

Historical remarks

On the basis of his studies of power series, Stifel (1487 - 1567) was one of the first who recognized the rules for logarithms as well as their usefulness. He was so enthusiastic about the saving in labour that he wrote: "One could publish an entire book about the wonderful properties of these numbers."

There remained one problem, which had to be overcome, namely to obtain the values of logarithms. The first who undertook this immense task was the Swiss mathematician Jobst Bürgi (1552 - 1632) with extraordinary mathematical talent and technical ability. He published his table only after long persuasion by his friend Kepler in 1620, so that the Scot Napier preceded him with his table published in 1614. The latter also coined the word logarithm, a combination of the Greek word logos (proportion) and arithmos (number). The English mathematician Henry Briggs (1556 - 1630) soon recognized the deficiency of the table of his friend Napier. They discovered that is would be best to start from log₁₀10 = 1. With true enthusiasm, he presented for an extensive range of numbers the first table of the logarithms to base 10, which today are referred to as Brigg's logarithms.

With the aid of power series, one can today find logarithms with any accuracy, a subject which we will discuss here.

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