Which equation represents an exponential function that passes through the point \( (2,80) \) ? \( f(x)=4(x)^{5} \) \( f(x)=5(x)^{4} \) \( f(x)=4(5)^{x} \) \( f(x)=5(4)^{x} \)

To identify the exponential function that passes through the point \( (2,80) \), we can substitute \( x = 2 \) into each equation and see which one yields \( f(2) = 80 \). 1. For \( f(x) = 4(5)^{x} \): \[ f(2) = 4(5)^{2} = 4 \times 25 = 100 \quad (\text{not } 80) \] 2. For \( f(x) = 5(4)^{x} \): \[ f(2) = 5(4)^{2} = 5 \times 16 = 80 \quad (\text{correct choice}) \] The equation \( f(x) = 5(4)^{x} \) represents the exponential function that passes through the point \( (2,80) \). Throughout history, exponential functions have helped in modeling various phenomena like population growth and radioactivity decay. Mathematician Jacob Bernoulli was one of the first to study compound interest scenarios, leading to the natural logarithm, which is a key to understanding growth rates. In real-world applications, exponential functions are everywhere! From calculating compound interest in finance, to measuring the spread of diseases in epidemiology, these functions provide insights into predictability and the potential for dramatic growth or decline. Always check your initial values and consider the context of the problem to avoid common pitfalls!