## Calculus without limits 5: log and exp

I’ve been on a bit of a logarithms kick lately, and I should say I’m not the only one. HowardAt58 has had a good number of articles about it, too, and I wanted to point some out to you. In this particular reblogging he brings a bit of calculus to show why the logarithm of the product of two numbere has to be the sum of the logarithms of the two separate numbers, in a way that’s more rigorous (if you’re comfortable with freshman calculus) than just writing down a couple examples along the lines of how 10

^{2}times 10^{3}is equal to 10^{5}. (I won’t argue that having a couple specific examples might be better at communicating the point, but there’s a difference between believing something is so and being able to prove that it’s true.)

The derivative of the log function can be investigated informally, as log(x) is seen as the inverse of the exponential function, written here as exp(x). The exponential function appears naturally from numbers raised to varying powers, but formal definitions of the exponential function are difficult to achieve. For example, what exactly is the meaning of exp(pi) or exp(root(2)).

So we look at the log function:-

## howardat58 8:41 pm

onThursday, 23 October, 2014 Permalink |Hey thanks, Joseph.

I quite agree with your comment about examples. Yes, examples first, then you know why you are trying to prove something generally.

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## Joseph Nebus 4:13 am

onWednesday, 29 October, 2014 Permalink |Thank you. I’ve wondered sometimes if the whole trick to teaching mathematics is figuring out just enough examples to set up the generalization, and then a couple examples to show why the generalization works. It’s surely got to be more than that, though.

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## nebusresearch | My Math Blog Statistics, October 2014 2:45 pm

onSaturday, 1 November, 2014 Permalink |[…] Calculus without limits 5: log and exp, a bit of reblogged work from HowardAt58 that gets into real analysis. […]

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