Ln x/y = ln x - ln y From Integral 1/x dx only ?

[morrobay]

Is the derivation of this relationship ln x/y = ln x - ln y
exclusively from, and originating from the evaluation of the Integral from 1 --> x 1/t dt = ln x ?
To say another way can ln x/y = ln x - ln y always be considered to apply to this integral ?

 

[Mentallic]

Not really, there are other ways to show it's true, such as using basic exponential formulae:

$$e^{x-y}=\frac{e^x}{e^y}$$

 

[AlephZero]

Yes you can show it direct from the integral.
$$\int_1^x\frac{dt}t - \int_1^y\frac{dt}t = \int_y^x\frac{dt}t$$
Then substitute u = t/y.

 

[morrobay]

Yes you can show it direct from the integral.
$$\int_1^x\frac{dt}t - \int_1^y\frac{dt}t = \int_y^x\frac{dt}t$$
Then substitute u = t/y.

Thanks to both of you . I am not questioning the validity of the correspondence to
the integral: 1 to x 1/x dx = ln x - ln 1
Rather confirmation that this correspondence is one to one .
That ln x is derived from the above integral alone.

 

[Char. Limit]

I'm really not sure what you mean. Do you mean that ln(x) is the only function f(x) that satisfies the property f(x/y) = f(x) - f(y)? Or do you mean that the integral is the only way of proving this property? Or perhaps something else entirely?

 

[Office_Shredder]

I think he means don't use the fact that you know ln(x) is an inverse of e^x

 

[morrobay]

Im trying to show that this property : ln x/y = ln x - ln y
is derived from the integral 1 to x 1/x dx

see this http://www.eoht.info/page/S+=+k+ln+W
I want to show that the natural logarithm in the statistical mechanics formulation
of entropy change , delta S = k ln W₂/W₁
is based on the natural logarithm for thermodynamic isothermal gas expansion:
delta S = nR integral V₁ to V₂ dv/V = ln V2/V1
That is based on the integral 1 to x 1/x dx = ln x - ln 1

 

[6.28318531]

You can prove ln(x/y) = ln(x)-ln(y), its similar to proving ln(xy)=ln(x)+ln(y), but you would use a different substitution. Start with the integral between 1 and x/y, and then split it into two integrals one between x and 1 AND the other between x/y and x. Then for the second integral use the substitution t=u/x,