How is the Logarithm derived from the exponential function? (aren't they inverses?)

Question

I've been learning logs in school, and my teacher, friend, and I are stumped on something. How does one derive the logarithmic function from the exponential function? My friend thinks Tayler Series are the trick. Is he right? Is there a a better/simpler/more elegant way? Also, do calculators use taylor series to do logs? Thanks for the help

Answer 1

The answer (at least, one possible answer) is in your title! You can define logarithms as inverses of exponential functions.

However, this then prompts the question: how do you define the exponential function? Again there are various ways in which you could do this. One common way is to say that the exponential function $f(x)=e^x$ is the unique function which has the properties $$\frac{d}{dx}(e^x)=e^x\quad\text{and}\quad e^0=1\ .$$ However, this raises some questions which are usually not answered (or worse, not even asked) in basic calculus courses. Here are two:

(1) How do we know that functions of the form $a^x$ are differentiable? After all, you will have met functions such as the absolute value which are not differentiable.

(2) Even if we assume that $a^x$ is differentiable, how do we know there is any value of $a$ which makes its derivative the same function? After all, this is just asking us to find $a$ by solving an equation, and there are many equations which have no solution, for example, $a=a+1$.

For these and other reasons it is often found better to do things the other way around: define the (natural) logarithm first by $$\ln x=\int_1^x\frac{dt}{t}$$ for $x>0$, and then define $e^x$ to be the inverse of $\ln x$.

It's a great question to think about and I hope this gives you a useful start.

A related question, also well worth thinking about: it's easy to say what we mean by $\pi^2$, but what exactly do we mean by $2^\pi\,$?

Answer 2

You define the natural logarithm as the inverse of the exponential function.

As for calculating logarithms, you can use taylor series or CORDIC like methods among other things - see this thread and this one.

Answer 3

Well, Taylor series expansions are certainly one way to derive the relationship between the exponential function and the natural logarithm.You can abstractly define the logarithm as the inverse of the exponential, but I think you're asking about a formula or calculational approach. A careful proof using calculus is not so easy as it requires careful definitions via limits. A very good discussion of this can be found at http://www.math.umn.edu/~jodeit/course/LogIn.pdf. There are a frieght car load of other proofs and derivations to be found both online and in standard textbooks, but I really like this one as it's much more meticulous then the standard calculus proofs.

Hope that answered your question!