Solve each application. The table shows the cost of a year's...

Solve each application. The table shows the cost of a year's tuition, room and board, and fees at a public university from $2000-2008$. Letting $y$ represent the cost and $x$ represent the number of years since $2000,$ we find that the function defined by $$f(x)=8160(1.06)^{x}$$ models the data quite well. According to this function, when will the cost in 2000 be doubled? $$\begin{array}{|c|c|}\hline\text { Year } & \text { Average Annual Cost } \\\hline 2000 & \$ 7,990 \\\hline 2001 & \$ 8,470 \\\hline 2002 & \$ 9,338 \\\hline 2003 & \$ 9,805 \\\hline 2004 & \$ 10,295 \\\hline 2005 & \$ 10,810 \\\hline 2006 & \$ 11,351 \\\hline 2007 & \$ 11,918 \\\hline 2008 & \$ 12,514 \\\hline\end{array}$$

Solve each application.
The table shows the cost of a year's tuition, room and board, and fees at a public university from $2000-2008$. Letting $y$ represent the cost and $x$ represent the number of years since $2000,$ we find that the function defined by $$f(x)=8160(1.06)^{x}$$ models the data quite well. According to this function, when will the cost in 2000 be doubled?
$$\begin{array}{|c|c|}\hline\text { Year } & \text { Average Annual Cost } \\\hline 2000 & \$ 7,990 \\\hline 2001 & \$ 8,470 \\\hline 2002 & \$ 9,338 \\\hline 2003 & \$ 9,805 \\\hline 2004 & \$ 10,295 \\\hline 2005 & \$ 10,810 \\\hline 2006 & \$ 11,351 \\\hline 2007 & \$ 11,918 \\\hline 2008 & \$ 12,514 \\\hline\end{array}$$

Key Concepts

Doubling Time

Doubling time is the specific period required for a quantity experiencing exponential growth to double in value. It is calculated by setting the exponential model equal to twice the initial value and solving for the time variable. This concept is widely applicable in contexts involving financial growth, population studies, and other areas where rapid increase is modeled.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are essential for solving equations where the variable is an exponent. In problems involving exponential growth, logarithms allow us to isolate the time variable by translating multiplicative relationships into additive ones. This technique is crucial when finding the exact time at which a quantity reaches a specific multiple of its initial value.

Exponential Growth

Exponential growth describes a process in which a quantity increases by a fixed percentage over equal time intervals. It is characterized by the function f(x) = a * b^x, where a is the initial value and b is the growth factor. This concept is key in modeling phenomena where compounding plays a central role, such as tuition increases, population growth, or radioactive decay.

Solving Exponential Equations

Solving exponential equations involves finding the value of the variable that appears in the exponent. This typically requires taking the logarithm of both sides of the equation to convert the exponential form into a linear one in terms of the variable. This method is fundamental in many applications, including determining doubling time and analyzing growth processes in various fields.

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