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In his Calculus book - A First Course in Calculus - Lang introduces the exponential function as enter image description here

Does Property 4 Assume the existence of such a function? On what basis? The illustration before the Property 4 seems to illustrate only at x=0. My question is specifically the way Lang introduces the function since I am following the book currently.

sidharth
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  • Yes, if that is what the text indicates – sidharth Mar 28 '19 at 05:48
  • If you have a definition of $a^b, a>0,b\in\mathbb {R} $ without use of $e^x, \log x$ then you can prove property $4$. An outline is available in my answer https://math.stackexchange.com/a/3162673/72031 It also contains link to my blog where all the details are provided. – Paramanand Singh Mar 28 '19 at 06:14
  • Another approach is to show that there exists a unique solution of $dy/dx=y$ with $y(0)=1$. See https://math.stackexchange.com/a/1292586/72031 and https://math.stackexchange.com/a/3155496/72031 – Paramanand Singh Mar 28 '19 at 06:16
  • Thanks for the links. Since I was following Lang, I wanted to understand the context in which he is developing the argument. The links seem helpful and I will explore them in detail. Many thanks – sidharth Mar 28 '19 at 06:27

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