Finding an Exponential Function In Exercises 23-26, find the exponential function y=C e^t t that

step 1 All right, so this is going to become a system of equations because we have a general formula an

Finding an Exponential Function In Exercises 23-26, find the...

Key Concepts

Exponential Function

An exponential function is a mathematical expression in the form y = C * e^(kt) where e is the base of the natural logarithm, C is the initial value or scaling constant, and k is the rate of growth (if k > 0) or decay (if k < 0). This type of function is commonly used to model processes that change at a rate proportional to their current value, such as population growth, radioactive decay, or interest accumulation.

Parameter Determination

Finding the specific exponential function that fits a given situation typically involves determining the values of the constants C and k. Using the coordinates of points that lie on the graph of the function, one can set up equations to solve for these parameters. The value of C is often directly given by the y-intercept when t = 0, while k can be determined by taking the ratio of the function’s values at two different points and applying logarithms.

System of Equations

When an exponential function is required to pass through two given points, the problem naturally leads to a system of two equations in two unknowns (usually C and k). Each point provides an equation by substituting into y = C * e^(kt). The system of equations can then be solved using algebraic techniques such as division (to eliminate one of the variables) followed by the application of logarithms to isolate and solve for the remaining parameter.

Logarithms

Logarithms are essential for solving equations involving exponents. When an exponential equation is transformed by taking the natural logarithm of both sides, the exponent is brought down as a multiplier. This property is key to isolating the parameter k in equations derived from exponential functions, thereby simplifying the process of finding the exact exponential representation that fits the given data points.

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