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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

Colin Michael Flaherty
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  • It is easy to show that the coefficients of a Taylor series that converges are its derivatives at $0$, and hence any function $f$ that satisfies $f'(x) = f(x)$ and has a Taylor series must have equal coefficients, and then check that such a Taylor series does converge, proving both existence and uniqueness up to a constant factor. While Taylor series may be used in calculators or fixed precision arithmetic, much faster arbitrary precision arithmetic algorithms for log include http://www.math.ust.hk/~machiang/education/enhancement/arithmetic_geometric.pdf. – user21820 Feb 21 '14 at 05:01
  • Alternatively, you can define exponentiation using integers first and extending to rationals and then irrationals, and show that the properties carry over to logarithms as I do at http://math.stackexchange.com/questions/620955/power-rule-for-logarithms/620961#620961. Then the special function involving $e$ becomes a question of which base is special. – user21820 Feb 21 '14 at 05:10

3 Answers3

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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.

Community
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Batman
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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 $2^\pi\,$?

David
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  • This is how I learned the relationship in calculus,David-and I agree, it's a great approach for a beginner. I gave a more rigorous answer to the questioner in my response. But this is definitely a very good approach,particularly for students interested in calculus! – Mathemagician1234 Feb 21 '14 at 04:26
  • Thanks! This is really helpful; it seems I do not know enough about the derivative and integral to completely understand your answer though. :( Would you say that once I understand these two ideas more, I will understand your answer more? Also, I'm not in calculus yet; do you happen to know of any good resources to learn more about it? Thanks for the help. @David – Colin Michael Flaherty Feb 21 '14 at 04:53
  • I should say that once you have done some calculus up to differentiation and basic integration you should be able to understand all this well. I teach this sort of stuff to students in a first year university calculus course. You might like to have a browse around Mathworld, starting with this. – David Feb 21 '14 at 05:06
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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!

Mathemagician1234
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