The table shows the cost of a year's tuition, room and...

The table shows the cost of a year's tuition, room and board, and fees at 4-year public colleges for the years $2006-2014$. Letting $y$ represent the cost in dollars and $x$ the number of years since $2006,$ the function $$f(x)=13,017(1.05)^{x}$$ models the data quite well. According to this function, in what year will the 2006 cost be doubled? $$\begin{array}{|c|c|} \hline \text { Year } & \text { Average Annual cost } \\ \hline 2006 & \$ 12,837 \\ \hline 2007 & \$ 13,558 \\ \hline 2008 & \$ 14,372 \\ \hline 2009 & \$ 15,235 \\ \hline 2010 & \$ 16,178 \\ \hline 2011 & \$ 17,156 \\ \hline 2012 & \$ 17,817 \\ \hline 2013 & \$ 18,383 \\ \hline 2014 & \$ 18,943 \\ \hline \end{array}$$

The table shows the cost of a year's tuition, room and board, and fees at 4-year public colleges for the years $2006-2014$. Letting $y$ represent the cost in dollars and $x$ the number of years since $2006,$ the function
$$f(x)=13,017(1.05)^{x}$$
models the data quite well. According to this function, in what year will the 2006 cost be doubled?
$$\begin{array}{|c|c|}
\hline \text { Year } & \text { Average Annual cost } \\
\hline 2006 & \$ 12,837 \\
\hline 2007 & \$ 13,558 \\
\hline 2008 & \$ 14,372 \\
\hline 2009 & \$ 15,235 \\
\hline 2010 & \$ 16,178 \\
\hline 2011 & \$ 17,156 \\
\hline 2012 & \$ 17,817 \\
\hline 2013 & \$ 18,383 \\
\hline 2014 & \$ 18,943 \\
\hline
\end{array}$$

Key Concepts

Exponential Growth Functions

An exponential growth function models a situation where the rate of increase of a quantity is proportional to its current value. This type of function is expressed in the form y = a·b^x, where a is the initial value, b is the growth factor, and x represents time. It is particularly useful for representing processes like population growth, compound interest, and in this case, the increase in college costs over time.

Doubling Time

Doubling time refers to the period required for a quantity undergoing exponential growth to double in size or value. In an exponential growth function, the doubling time can be determined mathematically through logarithms by solving the equation b^x = 2, where b is the growth factor. This concept is vital in understanding how quickly values increase over a given period.

Logarithms in Exponential Equations

Logarithms are the inverse operations of exponentiation and are used to solve for the exponent in exponential equations. When given an equation of the form b^x = k, taking the logarithm of both sides allows one to isolate x and determine its value. This method is essential for finding the time period in doubling time calculations and other problems involving exponential functions.

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