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It would really nice to have pictorial representation of how these functions are related and how each unknown can be derived. (A2A)

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    3Blue1Brown put out a video called [Triangle of Power](https://www.youtube.com/watch?v=sULa9Lc4pck) that answers this question almost perfectly. – 2017-01-29
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    I have seen his video. This was a A2A style question. However, meant to deduce these relation when one finds it confusing. Say drawing this on paper. As i understand , he intents a new notation i suppose ? – 2017-01-29

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enter image description here

There is a common relation between exponent, log and root, to know more about the intuition between these relation refer here. A simple trick to have all these relation derived is by just imagining this star.

This is meant for people with dyslexia, usually they get confused about these relation. Since we are really good with photo memory, this approach will help deduce these relations even in mind.

3 binary operators on the vertices of top triangle and 3 operands of bottom

  • ^ for the exponent nth power
  • $\sqrt[n]{}$ for the nth root
  • ${\log_n }$ log of nth base
  • b
  • x
  • y

Steps to find

  • Focus at the unknown variable of the context from the vertex of bottom triangle.

  • Follow the arrow at the the vertex of variable to the operator pointed by the arrow.

  • Apply the operation, while taking the operands in the clock wise order.

enter image description here

Finding x = b^y

Look at x , follow the arrow to the operator in the top triangle here ^ now apply the operation while taking the operands in the clock wise order so b^y

Finding y = ${\log_b x}$

Look at y, follow the arrow to the ${\log }$ (ie operator opposite to y), now apply the operands in clock wise order to get ${\log_b x}$

Finding b = $\sqrt[y]{x}$

left as an exercise to the reader.

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    Interesting, but to be honest this only confuses me more than anything else. – 2017-01-29
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    can you please shed light into what exactly is confusing. This is not a notation, however just quicker way to find how each unknowns in the context are related. Is it the star diagram that is confusing ? – 2017-01-29
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    Personally I learned all this stuff a long time ago. For me the easiest way to understand log$(z) = x$ is just $z = e^x$, and likewise for $n$th roots. I feel like remembering all the relationships encoded by this triangle is at least as complicated as understanding all the functions the traditional way, by their definitions. I do not see why it is natural at all. The same goes for the 3Blue1Brown video-- I have no idea why the triangle he suggests simplifies anything. I can see why it would be useful for dyslexic students, though. – 2017-01-29
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    Could you clarify this answer? I really don't understand what you mean by "Look at y, follow the allow to the log now apply the operands in clock wise order". – 2017-01-29
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    @AntonioVargas i have corrected it. Keep in mind this is not a notation, but simple diagram, so even though we say log to the base b of x. The clock wise order of how operands are taken ie from x to b may no comply to that statement. So consider this as memory aid. :). That is one of the main reason why i named the variable base to "b". – 2017-01-29
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    Frankly I still don't understand, but if this helps you as a memory aid then that's great. – 2017-01-30