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Advanced Math / Nonlinear functions Difficulty: Hard

$x$	$f of x$
$1$	$a$
$2$	$a, to the fifth power.$
$3$	$a, to the ninth power$

For the exponential function f, the table above shows several values of x and their corresponding values of $f of x$ , where a is a constant greater than 1. If k is a constant and $f of k, equals a, to the twenty ninth power$ , what is the value of k ?

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Explanation

The correct answer is 8. The values of $f of x$ for the exponential function f shown in the table increase by a factor of $a, to the fourth power$ for each increase of 1 in x. This relationship can be represented by the equation $f of x equals, a, raised to the 4 x plus b power$ , where b is a constant. It’s given that when $x equals 2$ , $f of x equals, a, to the fifth power$ . Substituting 2 for x and $a, to the fifth power$ for $f of x$ into $f of x equals, a, raised to the 4 x plus b power$ yields $a, to the fifth power equals, a, raised to the 4 times 2, plus b power$ . Since $4 times 2, plus b, equals 5$ , it follows that $b equals negative 3$ . Thus, an equation that defines the function f is $f of x equals, a, raised to the 4 x minus 3 power$ . It follows that the value of k such that $f of k equals, a, to the twenty ninth power$ can be found by solving the equation $4 k minus 3, equals 29$ , which yields $k equals 8$ .