To determine the multiplicative rate of change for the exponential function [tex]\( f(x) = 2 \cdot 5^x \)[/tex], we need to understand the general form of an exponential function, which is given by [tex]\( a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is a constant and [tex]\( b \)[/tex] is the base of the exponential function.
For the function [tex]\( f(x) = 2 \cdot 5^x \)[/tex]:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 5 \)[/tex]
The multiplicative rate of change in an exponential function is characterized by the base of the exponent, which is the factor by which the function values increase as [tex]\( x \)[/tex] increases by 1.
In our function [tex]\( f(x) = 2 \cdot 5^x \)[/tex], the base [tex]\( b \)[/tex] is 5. Thus, every time [tex]\( x \)[/tex] increases by 1, the value of [tex]\( f(x) \)[/tex] is multiplied by 5.
Therefore, the multiplicative rate of change of the function [tex]\( f(x) = 2 \cdot 5^x \)[/tex] is:
[tex]\[ \boxed{5} \][/tex]
