Exponential and power functions

Exponential functions, power functions and logarithmic functions are closely related, because they are all based on the rules of exponents.

 

An exponential function has the variable in the exponent, a power function has the variable as the base with a fixed exponent, and the logarithmic function is the inverse of the exponential function.

 

Exponential function:

 

$$ \Large f(x) = b \cdot a^x, \quad a>0, a \neq 1 $$

 

Power function:

 

$$ \Large f(x) = k \cdot x^a, \quad x>0 $$

 

Logarithmic function:

 

$$ \Large f(x) = \log_a(x), \quad a>0, a \neq 1 $$

 

Together these three types of functions form an important part of mathematics and are often used in growth models, physics and natural sciences.