Do you know of any Calculus text defining the exponential and logarithm functions in an alternative way?

Solution 1:

I like the approach Lang takes in Undergraduate Analysis. He defines the exponential as a function that satisfies the following differential equation subject to specified initial conditions:

$$ f^{\prime} = f, \;f(0)=1 $$

Using these assumptions he shows that if $f$ exists then it is unique. Later in the text he proves existence with power series. He gives an analagous treatment for $sin(x)$ and $cos(x)$. Fitzpatrick's Advanced Calculus takes a similar approach

Solution 2:

Maybe you need the G.M. Fikhtengolts's book A course in differential and integral calculus (Фихтенгольц Г.М.: Курс дифференциального и интегрального исчисления)? The vol.1 give a construction from the irrational power.

Solution 3:

I have seen other ways to do it. I know at least one analysis course at my university (for bio-engineers) defines the exponential function through its Taylor series and works its way from there.

The other approach I've seen extends from rational to real exponents by using Cauchy sequences of rational numbers.

I must say neither of them are basic calculus texts, but they are still basic analysis texts.