Question

Write an exponential decay function to model each situation. Then estimate the value of x for the given value of f(x)

initial value: 100
decay factor: .95
f(x)=60

Answer 1

• f(x) = 100(0.95^x)
• x ≈ 9.959 for f(x) = 60

Explanation:

You want the value of x that makes f(x) = 60, when f(x) is the exponential function with an initial value of 100 and a decay factor of 0.95.

Exponential function

The form of an exponential function is ...

f(x) = (initial value) · (decay factor)^x

Application

For the given initial value and decay factor, the function is ...

f(x) = 100 (0.95^x)

Solving for x, we find ...

f(x)/100 = 0.95^x

log(f(x)/100) = x·log(0.95)

x = log(f(x)/100)/log(0.95)

For f(x) = 60, the value of x is about ...

x = log(60/100)/log(0.95) ≈ 9.959

The value of x is about 9.959.