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If we have

$$ x^y = z $$

then we know that

$$ \sqrt[y]{z} = x $$

and

$$ \log_x{z} = y .$$

As a visually-oriented person I have often been dismayed that the symbols for these three operators look nothing like one another, even though they all tell us something about the same relationship between three values.

Has anybody ever proposed a new notation that unifies the visual representation of exponents, roots, and logs to make the relationship between them more clear? If you don't know of such a proposal, feel free to answer with your own idea.

This question is out of pure curiosity and has no practical purpose, although I do think (just IMHO) that a "unified" notation would make these concepts easier to teach.


  • 0
    (BTW, is it just me or is the TeX preview on the question form not working today?)2011-03-31
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    My TeX preview hasn't been working either. I don't understand your question though, what is wrong with the current notation?2011-03-31
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    There's nothing wrong with it, I just think it's inelegant to have three symbols that are so different to describe three parts of the same relationship. I think it would be helpful for learners to see the relationship between logs and roots visually.2011-03-31
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    Be careful about saying that these three statements are equivalent when the corresponding functions aren't always well-defined for all $x, y, z$... restricting to positive reals makes everything okay, though.2012-07-03
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    Although you wouldn't restrict $y$; just $x$ and $z$.2012-07-04
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    This whole program is totally misguided. There are three different symbols because there are three qualitatively different functions. To have analogous notation for the logarithmic and exponential functions - e.g. by using a triangle with 3 seemingly symmetric vertices - would be as actively harmful as to have similar words for "giving the birth" and "murdering". Also, the natural elementary functions are just ln(x) and exp(x) which only have one argument, not two, and the triangle-style notation further prevents people from understanding why e=2.718... is the most natural base.2016-07-19
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    @LubošMotl I'm not sure that the current notation helps that, either. You can throw around $\exp x$ and $\ln x$ all you want, but until you explain derivatives of exponentials and the convergence of exponential growth, it's not going to make much sense anyway. Picture a 5th grader learning about exponents and radicals, or an eighth grader learning about logarithms. How would you explain to them why $e$ is a much better base than $10$?2018-09-14
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    OK, I understood it as a third-grader. It has nothing to do with age. It's about the availability of the lessons. $e$ is explained to kids using the interest - which I first saw in a science journal for kids, VTM. Start with \$1, add 100% interest, you have \$2. Instead, add 50% twice, you get \$2.25 (1.5 times 1.5). Add 100 times 1%, you will get about \$2.7. There's a finite limit of $(1+1/N)^N$ and this number 2.718 is the most natural base - it's the coefficient how much something grows continuously in geometric series with the most natural finite growth rate. No derivatives needed.2018-09-16
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    But even if those things were only explained at the high school, they're still true. If you teach the kids to use a conceptually misguided notation, it will prevent them from understanding these things at the high school - which is still a serious enough problem. What's actually going on is that some of the folks don't understand $e$ and why it's more natural even as adults, and these people would like to determine education or mathematical notation. That's a path to eliminate mathematically literate kids from schools and from the future of nations.2018-09-16

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