Eeponentel and togerthmic Functons
Finding the time to reach a limit in a word problem on exponential growt.

A laptop computer is purchased for $\$ 2500$. Each year, its value is $75 \%$ of its value the year before. After how many years will the laptop computer be worth $\$ 500$ or less? (Use the calculator provided if necessary.)

Write the smallest possible whole number answer.
Џyears

Step 1 ：Translate the problem into a mathematical model. This is an exponential decay problem, where the value of the laptop decreases by a certain percentage each year. We can model this situation with the exponential decay formula: \(V = P * (1 - r)^t\), where \(V\) is the final value ($500), \(P\) is the initial value ($2500), \(r\) is the rate of decrease (25% or 0.25), and \(t\) is the time in years.

Step 2 ：We want to find the smallest \(t\) such that \(V <= $500\). We can rearrange the formula to solve for \(t\): \(t = \log_{(1 - r)}\frac{V}{P}\).

Step 3 ：Plug in the given values and solve for \(t\): \(P = 2500\), \(V = 500\), \(r = 0.25\), \(t = 6\).

Step 4 ：Final Answer: The laptop computer will be worth $500 or less after \(\boxed{6}\) years.