Exponential and Logarithmic Functions
Evaluating an exponential function that models a real-world situation

If the rate of inflation is $2.6 \%$ per year, the future price $p(t)$ (in dollars) of a certain item can be modeled by the foll the number of years from today.
\[
p(t)=2500(1.026)^{t}
\]

Find the current price of the item and the price 10 years from today.
Round your answers to the nearest dollar as necessary.
Current price:
Price 10 years from today: $\$ \square$

Step 1 ：The current price of the item is given by the function when t = 0. We know that any number raised to the power of 0 is 1, so: \(p(0) = 2500 * (1.026)^0 = 2500 * 1 = 2500\). So, the current price of the item is $2500.

Step 2 ：To find the price 10 years from today, we substitute t = 10 into the function: \(p(10) = 2500 * (1.026)^{10}\).

Step 3 ：To calculate this, we first calculate the value of \((1.026)^{10}\), and then multiply the result by 2500. \((1.026)^{10} ≈ 2.709\).

Step 4 ：So, \(p(10) = 2500 * 2.709 ≈ 6773\).

Step 5 ：So, the price of the item 10 years from today will be approximately $6773, rounded to the nearest dollar. \(\boxed{6773}\)