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- Thread starter Ralphz
- Start date

- Joined
- Aug 27, 2012

- Messages
- 1,212

Let's use base e and just call it "log."

[math]log(a) = c ~ log(b)[/math]

[math]c = \dfrac{log(a)}{log(b)} = \dfrac{ \text{constant} }{log(b)}[/math]

so c as a function of b is essentially just 1 over log. Nothing new.

-Dan

Addendum: Notation comment. What you wrote "a = b(exp)c" is written in text as "a = b^c"

- Joined
- Jun 18, 2007

- Messages
- 25,213

what is the function in which the base is the dependent variable and the exponent then varies continuously as the argument is held constant.

y

y = [c]

y = c

This is called an implicit function. If a is a positive constant

[MATH]a =x^y \implies x = a^{1/y} \text { AND }y = a * \dfrac{ln(a)}{ln(x)} \text { unless } a = 1.[/MATH]

It is your choice.

The above is one of 3 potential pairs of variables; the other 2 represent the power and exponential functions, both deeply central to mathematics ordained as such by their respective names. What do we understand of the role of the 3rd pairing? What are its correspondences in nature and to deeper math? Or perhaps it is just the odd man out?