defining the natural logarithm

[richardt]

Greetings:

Why is it that we define ln(x) as ∫(1/t) dt ; (t₁, t₂) = (1, x), rather than Ln(x) = y if and only if e^y = x; x > 0 ? I understand that the integral does indeed have value ln(x) and that such definition is therefore valid. Is there some advantage to this definition? Or is it simply intended to emphasize the "natural" nature of the log base e?

Thank you kindly.

Rich B.

 

[pka]

Why is it that we define ln(x) as ∫(1/t) dt ; (t₁, t₂) = (1, x), rather than Ln(x) = y if and only if e^y = x; x > 0 ? I understand that the integral does indeed have value ln(x) and that such definition is therefore valid. Is there some advantage to this definition? Or is it simply intended to emphasize the "natural" nature of the log base e?

I cannot prove this as the source, but I first learned this definition from Leonard Gillman.

Actually in his textbook he defined \(\displaystyle \displaystyle\left( {\forall x > 0} \right)\left[ {L(x) = \int_1^x {{t^{ - 1}}dt} } \right]\).

Then using that function we can easily prove that \(\displaystyle L(1) = 0~\&~L(xy)=L(x)+L(y)\),
the two essential properties of a logarithm function. It is also a natural definition.

I also agree with Gillman that we should use \(\displaystyle \log(x)\) everywhere \(\displaystyle \ln(x)\) is now used.

 

[JeffM]

Greetings:

Why is it that we define ln(x) as ∫(1/t) dt ; (t₁, t₂) = (1, x), rather than Ln(x) = y if and only if e^y = x; x > 0 ? I understand that the integral does indeed have value ln(x) and that such definition is therefore valid. Is there some advantage to this definition? Or is it simply intended to emphasize the "natural" nature of the log base e?

Thank you kindly.

Rich B.

Logarithms to the base 10 were understood decades before calculus was invented. Look up Napier. I suspect Romesk is correct: e and log to the base e were discovered as part of the process of developing differential and integral calculus to fill in gaps in the power rule. If that historical surmise is true, they probably were not put on a firm theoretical basis until the last half of the 19th century.

In terms of what pka said about notation, logs to the base 10 were used for practical purposes at least through the 1960's, and a log to the base 10 was what was meant by log if no base was specified until at least the 1970's. If log to the base e was meant, ln was used. I personally see little advantage in changing notation because ln is more concise and completely unambiguous whereas log without an indicator of base is less concise and means different things depending on the date when it was used. But that is a purely personal idiosyncracy.