What is the effect of changing the base from 2 to 3 in the function [tex]f(x)=2^x[/tex] to [tex]f(x)=3^x[/tex]?
Choose the correct answer from the choices.
A. There is no change in the rate.
B. It decreases the growth rate.
C. It turns the function into a decay.
D. It increases the growth rate.
Answer (2)
Changing the base from 2 to 3 in the function increases the growth rate, meaning f ( x ) = 3 x grows faster than f ( x ) = 2 x . The correct answer is D. It increases the growth rate.
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Exponential functions have the form f ( x ) = b x , where b is the base.
When 1"> b > 1 , the function represents exponential growth.
Since 2"> 3 > 2 , f ( x ) = 3 x grows faster than f ( x ) = 2 x .
Changing the base from 2 to 3 increases the growth rate. The answer is It increases the growth rate.
Explanation
Understanding the Problem We are given two exponential functions, f ( x ) = 2 x and f ( x ) = 3 x . We want to determine the effect of changing the base from 2 to 3 on the growth rate of the function.
Understanding Exponential Functions The general form of an exponential function is f ( x ) = b x , where b is the base. The growth rate of the function is determined by the value of the base b . If 1"> b > 1 , the function represents exponential growth. If 0 < b < 1 , the function represents exponential decay.
Comparing the Growth Rates In our case, we are comparing f ( x ) = 2 x and f ( x ) = 3 x . Both have bases greater than 1, so they both represent exponential growth. Since 2"> 3 > 2 , the function f ( x ) = 3 x grows faster than f ( x ) = 2 x .
Conclusion Therefore, changing the base from 2 to 3 increases the growth rate.
Examples
Exponential functions are used to model population growth. If a population doubles every year, the growth can be modeled by f ( x ) = 2 x , where x is the number of years. If the population triples every year, the growth can be modeled by f ( x ) = 3 x . The higher the base, the faster the population grows. This concept is also applicable in finance, where compound interest can be modeled using exponential functions. A higher interest rate (analogous to a higher base) leads to faster growth of the investment.
