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the table could stop . The source of the problem was the value of log 10, because
    log 10 x = log 10 + log x
    Unfortunately log 10 was messy: with the base 1.0000000001 the logarithm of 10 was 23,025,850,929. Briggs thought it would be much nicer if the base could be chosen so that log 10 = 1. Then log 10 x = 1 + log x , so that whatever log 1.23456 might be, you just had to add 1 to it to get log 12.3456. Now tables of logarithms need only run from 1 to 10. If bigger numbers turned up, you just added the appropriate whole number.
    To make log 10 = 1, you do what Napier did, using a base of 1.0000000001, but then you divide every logarithm by that curious number 23,025,850,929. The resulting table consists of logarithms to base 10, which I’ll write as log 10 x . They satisfy
    log 10 xy = log 10 x + log 10 y
    as before, but also
    log 10 10 x = log 10 x + 1
    Within two years Napier was dead, so Briggs started work on a table of base-10 logarithms. In 1617 he published Logarithmorum Chilias Prima (‘Logarithms of the First Chiliad’), the logarithms of the integers from 1 to 1000 accurate to 14 decimal places. In 1624 he followed it up with Arithmetic Logarithmica (‘Arithmetic of Logarithms’), a table of base-10 logarithms of numbers from 1 to 20,000 and from 90,000 to 100,000, to the same accuracy. Others rapidly followed Briggs’s lead, filling in the largegap and developing auxiliary tables such as logarithms of trigonometric functions like log sin x .
    The same ideas that inspired logarithms allow us to define powers x a of a positive variable x for values of a that are not positive whole numbers. All we have to do is insist that our definitions must be consistent with the equation x a x b = x a + b , and follow our noses. To avoid nasty complications, it is best to assume x is positive, and to define x a so that this is also positive. (For negative x , it’s best to introduce complex numbers, as in Chapter 5 .)
    For example, what is x 0 ? Bearing in mind that x 1 = x , the formula says that x 0 must satisfy x 0 x = x 0+1 = x . Dividing by x we find that x = 1. Now what about x −1 ? Well, the formula says that x −1 x = x −1+1 = x 0 = 1. Dividing by x , we get x −1 = 1/ x . Similarly x −2 = 1/ x 2 , x −3 = 1/ x 3 , and so on.
    It starts to get more interesting, and potentially very useful, when we think about x 1/2 . This has to satisfy x 1/2 x 1/2 = x 1/2+1/2 = x 1 = x . So x 1/2 , multiplied by itself, is x . The only number with this property is the square root of x . So x 1/2 =. Similarly, x 1/3 =, the cube root. Continuing in this manner we can define x p/q for any fraction p/q . Then, using fractions to approximate real numbers, we can define x a for any real a . And the equation x a x b = x a+b still holds.
    It also follows that loglog x , and loglog x , so we can calculate square roots and cube roots easily using a table of logarithms. For example, to find the square root of a number we form its logarithm, divide by 2, and then work out which number has the result as its logarithm. For cube roots, do the same but divide by 3. Traditional methods for these problems were tedious and complicated. You can see why Napier showcased square and cube roots in the preface to his book.
    As soon as complete tables of logarithms were available, they became an indispensable tool for scientists, engineers, surveyors, and navigators. They saved time, they saved effort, and they increased the likelihood that the answer was correct. Early on, astronomy was a major beneficiary, because astronomers routinely needed to perform long and difficult calculations. The French mathematician and astronomer Pierre Simon de Laplace said that the invention of logarithms ‘reduces to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust’. As the use of machinery in manufacturing grew, engineers started to make more and more