How to intuitively deduce the relationship among exponent, log and root?

Question

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

Answer 1

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.

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.