How Can We Prove \( f(x) + f(y) = f(xy) \) Using Integral Definitions?

[psholtz]

If we define a function f(x) such that:

$$f(x) = \int_{1}^x \frac{dt}{t}$$

for $$x>0$$, so that:

$$f(y) = \int_{1}^y \frac{dt}{t}$$

and

$$f(xy) = \int_{1}^{xy} \frac{dt}{t}$$

is there a way, using just these "integral" definitions, to prove that:

$$f(x) + f(y) = f(xy)$$

Clearly, the function we are dealing w/ is the logarithm, but I'd like to prove this from the "definitions" given above, rather than reverting to "known properties" of the logarithm function.

 

[Mark44]

$$f(xy) = \int_{1}^{xy} \frac{dt}{t} = \int_1^x \frac{dt}{t} + \int_x^{xy} \frac{dt}{t}$$

By making a change to the dummy variable in the last integral, you can show it to be equal to f(y).

 

[psholtz]

I thought about doing a Taylor expansion about $$x=1$$, that is to say:

$$f(1+x) = f(1) + f'(1)x + \frac{1}{2!}f''(1)x^2 + \frac{1}{3!}f'''(1)x^3 + ...$$

$$f(1+x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - \frac{1}{4}x^4 + ...$$

but I'm not sure this leads anywhere fruitful.. First, we'd have to get an expression for $$f(x)$$, rather than $$f(1+x)$$:

$$f(x) = (x-1) - \frac{1}{2}\left(x-1\right)^2 + \frac{1}{3}\left(x-1\right)^3 - \frac{1}{4}\left(x-1\right)^4 + ...$$

but, if added to the equivalent expression for $$f(y)$$, I'm not sure this would lead to anything like an expression:

$$f(xy) = \left(xy-1\right) - \frac{1}{2}\left(xy-1\right)^2+ ...$$

 

[psholtz]

$$f(xy) = \int_{1}^{xy} \frac{dt}{t} = \int_1^x \frac{dt}{t} + \int_x^{xy} \frac{dt}{t}$$

By making a change to the dummy variable in the last integral, you can show it to be equal to f(y).

OK... yes. Thanks!

 

[Count Iblis]

Change of variables by rescaling t works. E.g. you can write:

$$f(x) = \int_{1}^{x}\frac{dt}{t} = \int_{\frac{1}{x}}^{1}\frac{du}{u} = - \int_{1}^{\frac{1}{x}}\frac{du}{u} = -f\left(\frac{1}{x}\right)$$

 

[psholtz]

Change of variables by rescaling t works. E.g. you can write:

$$f(x) = \int_{1}^{x}\frac{dt}{t} = \int_{\frac{1}{x}}^{1}\frac{du}{u} = - \int_{1}^{\frac{1}{x}}\frac{du}{u} = -f\left(\frac{1}{x}\right)$$

Which, in fact, is another useful property of the logarithm:

$$f(x) = -f\left(\frac{1}{x}\right)$$

 

[Ben Niehoff]

It was traditional in older calculus books to prove all the properties of the logarithm from the integral definition (or at least, they did in my mom's old calculus text she loaned me once). I think newer books may have decided this was "too confusing" or something.

As you can see, it's easy to show that the integral has the desired properties. You can also prove that it is the inverse function to $\exp x$.

 

[Landau]

It was traditional in older calculus books to prove all the properties of the logarithm from the integral definition (or at least, they did in my mom's old calculus text she loaned me once). I think newer books may have decided this was "too confusing" or something.

Spivak also does it.

 

[psholtz]

It was traditional in older calculus books to prove all the properties of the logarithm from the integral definition (or at least, they did in my mom's old calculus text she loaned me once). I think newer books may have decided this was "too confusing" or something.

As you can see, it's easy to show that the integral has the desired properties. You can also prove that it is the inverse function to $\exp x$.

Yes .. in point of fact, the question arose in my mind while I was reading one of the older calculus texts, specifically, Forsythe on Differential Equations. Forsythe mentions passingly that all the "properties" of the logarithm can be derived from its integral representation/definition, but doesn't go into any further detail.

Forgive my ignorance, but how can you derive that the exponential function is the inverse of the logarithm, given only the integral expression for logarithm?

 

[Ben Niehoff]

One way is this: We know that e^x solves the differential equation

$$\frac{dy}{dx} = y$$

Considering x as a function of y, we obtain

$$\frac{dx}{dy} = \frac{1}{y}$$

and hence the inverse function is

$$x(y) = \int_{y_0}^y \frac{dt}{t}$$

and it merely remains to fix the constant y_0.

Of course, the first differential equation is also solved by any constant multiple of e^x. With a little bit more algebra, you can take that into account.

 

[psholtz]

One way is this: We know that e^x solves the differential equation

$$\frac{dy}{dx} = y$$

Considering x as a function of y, we obtain

$$\frac{dx}{dy} = \frac{1}{y}$$

and hence the inverse function is

$$x(y) = \int_{y_0}^y \frac{dt}{t}$$

and it merely remains to fix the constant y_0.

Of course, the first differential equation is also solved by any constant multiple of e^x. With a little bit more algebra, you can take that into account.

Interesting, thanks.

Yes, that makes sense.

Also, if we don't yet "know" what the symbol "e" is supposed to mean, I suppose one could also proceed by supposing that the solution y is in the form of an (indeterminate) polynomial, in the sense of:

$$y(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + ...$$

so that from y'=y we get:

$$y'(x) = a_1 + 2a_2x + 3a_3x^2 + 4a_4x^3 + ...$$

and setting the constants equal, we get:

$$a_1 = a_0, a_2 = \frac{1}{2}a_1, a_3 = \frac{1}{3}a_2, a_4 = \frac{1}{4}a_3, ...$$

so that the solution works out to:

$$y(x) = a_0 + a_0x + \frac{1}{2!}a_0x^2 + \frac{1}{3!}a_0x^3 + ...$$

$$y(x) = a_0 \sum_{k=0}^{\infty} \frac{x^k}{k!}$$

which summation can be taken as the "definition" of $$e^x$$.

 

[The Chaz]

$$f(xy) = \int_{1}^{xy} \frac{dt}{t} = \int_1^x \frac{dt}{t} + \int_x^{xy} \frac{dt}{t}$$

By making a change to the dummy variable in the last integral, you can show it to be equal to f(y).

Could you please elaborate (or just do it outright!) ?
Thanks

 

[arildno]

Set u=t/x

Then, we have:
$$t=xu, \frac{dt}{du}=x\to{dt}=xdu$$
We therefore get:
$$\int_{x}^{xy}\frac{dt}{t}=\int_{1}^{y}\frac{xdu}{xu}=\int_{1}^{y}\frac{du}{u}=f(y)$$

 

[HallsofIvy]

It was traditional in older calculus books to prove all the properties of the logarithm from the integral definition (or at least, they did in my mom's old calculus text she loaned me once). I think newer books may have decided this was "too confusing" or something.

As you can see, it's easy to show that the integral has the desired properties. You can also prove that it is the inverse function to $\exp x$.

Oddly enough, I was thinking of this as how newer calculus books did it! I may just be showing my age.

 

[arildno]

Oddly enough, I was thinking of this as how newer calculus books did it! I may just be showing my age.

I thought so.

Back in your days, you had to make to do with tiny counting stones, right?

 

[HallsofIvy]

I thought so.

Back in your days, you had to make to do with tiny counting stones, right?

That is an insult! We used the abacus! (I knew the man who invented them.)

 

[HallsofIvy]

By the way, here is an important property of the logarithm not often shown in texts that define the logarithm in terms of the integral.

Since ln(x) is defined as an integral it is clearly differentiable for all positive numbers and, in particular, is differentiable on [1, 2] so we can apply the mean value theorem to that interval: there exist c in [1, 2] such that
$$\frac{ln(2)- ln(1)}{2- 1}= (ln)'(c)$$

Since
$$ln(x)= \int_1^t \frac{1}{x}dt$$
by the "fundamental theorem of caluculus" $(ln(x))'= 1/x$ for x positive.

Further
[tex]ln(1)= \int_1^1 \frac{1}{t}dt= 0[/itex]
because $\int_a^a f(t)dt= 0$ for any integrable function f.

So that gives
$$\frac{ln(2)- ln(1)}{2- 1}= \frac{ln(2)- 0}{1}= ln(2)= \frac{1}{c}$$
for some c between 1 and 2. But if $c\le 2$, then $1/c\ge 1/2$ so that $ln(2)\ge 1/2$!

Why is that important? Because if X is any positive number, then $ln(2^{2X})= 2X ln(2)\ge X$. That is, given any positive number there exist x so that ln(x) is larger than that: ln(x) does not have an upper bound. Since the derivative of ln(x) is 1/x> 0, ln(x) is an increasing function. Since there is no upper bound, we must have $\lim_{x\to\infty}ln(x)= \infty$. Since ln(1/x)= -ln(x), we also have $\lim_{x\to -\infty} ln(x)= -\infty$.

That is, ln(x) maps the set of positive real numbers one-to-one and onto the set of all real numbers. Because that is true, it has an inverse function, which we can call "Exp(x)".

Now, suppose y= Exp(x). Then x= ln(y). If $x\ne 0$, $1= (1/x)ln(y)= ln(y^{1/x})$. Now going back to the exponential form, $Exp(1)= y^{1/x}$ so that $y= (Exp(1))^x$. Of course, if x= 0, y= Exp(0)= 1 and $Exp(1)^0= 1$ so this is still true.

That is, from the integral definition of ln(x) we can recover the fact that it is the inverse function to some number to the x power.

 

[arildno]

That is an insult! We used the abacus! (I knew the man who invented them.)

 

[Bohrok]

How would you know to make the lower limit 1 when first defining ln x
$$\ln x = \int_1^x \frac{1}{t} dt$$?
Why not 2, e, or 10? Or is it like the same kind of magic sometimes used in limit proofs for picking delta?

 

[arildno]

If you pick 2, you will have a different function, that's all.

 

[The Chaz]

@Halls...
Double-check those limits. I think you stated in English something different than you wrote in symbols.

 

[Landau]

Since
$$ln(x)= \int_1^t \frac{1}{x}dt$$

To prevent confusing, this should of course read
$$\ln x = \int_1^x \frac{1}{t}dt$$

 

[Gib Z]

Why is that important? Because if X is any positive number, then $ln(2^{2X})= 2X ln(2)\ge X$. That is, given any positive number there exist x so that ln(x) is larger than that: ln(x) does not have an upper bound..

We can also establish this by comparison to the Harmonic Series.

 

[Bohrok]

If you pick 2, you will have a different function, that's all.

Ok, so can you still derive e^x easily from that function using 2 as the lower limit? I'd try it myself but I don't really know how to start it...

 

[HallsofIvy]

If you use "2" as a lower limit you get
[tex]f(x)= \int_2^x dt/t= \int_1^x dt/t- \int_1^2 dt/t= ln(x)- ln(2)[/itex]

Of course, that's now different function and has a different inverse. If y= f(x)= ln(x)- ln(2) then ln(x)= y+ ln(2) and so $x= e^{y+ ln(2)}= e^y e^{ln(2)}= 2e^x$ so $f^{-1}(x)= 2e^x$.