Assignment 7.4 Score: \( 3 / 100 \) Answered: \( 1 / 17 \) Question 2 Find the exponential function \( f(x)=a^{x} \) whose graph goes through the point \( \left(2, \frac{1}{25}\right) \). \( a= \) Question Help: Video Submit Question

To find the exponential function \( f(x) = a^x \) that passes through the point \( \left(2, \frac{1}{25}\right) \), we can plug in the values into the equation. This means we set up the equation: \[ \frac{1}{25} = a^2. \] Since \( \frac{1}{25} = 25^{-1} = (5^2)^{-1} = 5^{-2} \), we can rewrite the equation as: \[ a^2 = 5^{-2}. \] Taking the square root of both sides gives us \( a = 5^{-1} = \frac{1}{5} \). Thus, the exponential function is \( f(x) = \left(\frac{1}{5}\right)^x \). --- Before the invention of calculators, mathematicians relied heavily on logarithmic tables to deal with exponential functions. This method was crucial in performing calculations for various scientific and engineering applications. It's fascinating how these tables transformed complex multiplications into easy additions, a precursor to our modern-day calculator functions! In real-world scenarios, exponential functions appear in population growth, finance, and even in the spread of diseases. For instance, if a certain bacteria multiplies exponentially, knowing the growth rate can help predict its future population, leading to effective control measures in healthcare. So, whether it's planning for investments or managing health outbreaks, understanding exponential functions is key!