# A Third function

#### Ralphz

##### New member
For a=b(exp)c, fixing the b, or the base, yields the exponential function or its inverse, the logarithmic function; fixing the exponent, or c, yields the power function. So what function do you get when you fix a?

#### topsquark

##### Senior Member
$a = b^c$
Let's use base e and just call it "log."

$log(a) = c ~ log(b)$
$c = \dfrac{log(a)}{log(b)} = \dfrac{ \text{constant} }{log(b)}$
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"

#### Ralphz

##### New member
Thank you. I am think what happens when the base, b, varies continuously and the exponent varies continuously while the argument is held constant.

#### Ralphz

##### New member
To put it another way, what is the function in which the base is the dependent variable and the exponent then varies continuously as the argument is held constant. The inverse function would be the exponent as dependent variable as the base varies continuously with the argument constant.

#### Subhotosh Khan

##### Super Moderator
Staff member
what is the function in which the base is the dependent variable and the exponent then varies continuously as the argument is held constant.

yf1(x) = c

y = [c]{1/f1(x)}

y = cf(x)

#### JeffM

##### Elite Member
To put it another way, what is the function in which the base is the dependent variable and the exponent then varies continuously as the argument is held constant. The inverse function would be the exponent as dependent variable as the base varies continuously with the argument constant.
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]