Question

Y=−3(2.5)x

Does the equation represent exponential growth or decay? Choose the correct answer


Identify the rate of exponential growth or decay. Give your answer as a percentage.

Answer 1

1. The equation represent an exponential decay

2. The rate of the exponential decay is -3×2.5ˣ·㏑(2.5)

Explanation:

When a function a(t) = a₀(1 + r)ˣ has exponential growth, the logarithm of x grows with time such that;

log a(t) = log(a₀) + x·log(1 + r)

Hence in the equation -3 ≡ a₀, (1 + r) ≡ 2.5 and y ≡ a(t). Plugging in the values in the above equation for the condition of an exponential growth, we have;

log y = log(-3) + x·log(2.5)

Therefore, since log(-3) is complex, the equation does not represent an exponential growth hence the equation represents an exponential decay.

The rate of the exponential decay is given by the following equation;

`(dy)/(dx) =(d(-3(2.5)^x))/(dx) = -(d(3\cdot e^(x\cdot ln(2.5))))/(dx) = -3 * 2.5^x* ln(2.5)`

Hence the rate of exponential decay is -3×2.5ˣ × ㏑(2.5)