Logarithms and Their Infinite Series

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    Lay out the infinite series

(Eq地 1)

and integrate it with respect to x. We get

(Eq地 2)

What kind of function does F(1+x) represent?

    To answer that question let

(Eq地 3)

and differentiate it. We get

(Eq地 4)

Dividing that equation by Equation 3 and integrating the result gives us

(Eq地 5)

If we compare that equation with Equation 2 we find that we must write

(Eq地 6)

Also, letting x=0 in Equation 2 tells us that

(Eq地 7)

Combining Equations 6 and 7 tells us that

(Eq地 8)

which means that

(Eq地 9)

Equations 6, 7, and 9 display the fundamental properties of the natural (or Naperian) logarithm, so now we know that the statement

(Eq地 10)

stands true to mathematics.

    All of the above only stands true to mathematics with the proviso that x only take values between minus one and plus one. We need that proviso to ensure that the infinite series converge onto some finite value. But that proviso seems to restrict us to calculating logarithms only for numbers between zero and two. For obtaining the logarithms of larger numbers we must use a ploy suggested by Equation 6.

    Suppose I want to calculate the logarithm of 3.5 using the infinite series shown above. I need only calculate the logarithm of the product of 1.75 and 2 as the sum of the logarithms of 1.75 and 2. By using that ploy over and over, as many times as necessary, we can calculate the logarithm of any number at all without worrying about non-converging infinite series.

    We can integrate Equation 10 with respect to x by two different methods. We can integrate the infinite series of Equation 2 and get

(Eq地 11)

And we can integrate the logarithm by parts, exploiting the product rule for differentiation in the form

(Eq地 12)

In this case f=ln(1+x) and g=x. We get

(Eq地 13)

We can test that solution by replacing the logarithms with their infinite-series expansions. So

(Eq地 14)

which is what we got in Equation 11. In going from the second line in that equation to the third I exploited the fact that

(Eq地 15)

which is the p=1 manifestation of the general formula

(Eq地 16)

    Now let痴 look at a different series,

(Eq地 17)

That gives us an easy integration,

(Eq地 18)

That series looks like x plus a series for a logarithm, but does that impression give us an actual mathematical truth? Again we define a product, as in Equation 3, and differentiate it, so that we have

(Eq地 19)

Divide that equation by the product and integrate the result to get

(Eq地 20)

That equation shows that the integral on the left side of the equality sign has the fundamental property of a logarithm (ln(fg)=lnf + lng), but it痴 not the function that we want.

    We can relate it to the function that we want by bringing in Equation 17 so that we can integrate it explicitly. We get

(Eq地 21)

We have another way in which we can prove and verify the proposition that the integral on the left side of that equation is a true logarithm. We know that for any given function f(x) we have

(Eq地 22)

The integral on the left side of Equation 21 gives us

(Eq地 23)

We also know that x=ln(ex), so we can rewrite Equation 21 as

(Eq地 24)

which gives us

(Eq地 25)

    To check that result we exploit the fact that

(Eq地 26)

We can integrate that directly to get

(Eq地 27)

And we can also get the infinite series version,

(Eq地 28)

    Next we integrate Equation 21 and get

(Eq地 29)

There痴 no known closed-form equivalent of that infinite series, so we will have to devise one, either by finding some clever way to sum the series or by integrating the logarithm.

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