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Write an exponential function in the form
y=ab^(x) that goes through the points
(0,7) and
(3,3584).
Answer:

Write an exponential function in the form $y=a b^{x}$ that goes through the points $(0,7)$ and $(3,3584)$ . $\newline$ Answer:

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Convert between exponential and logarithmic form: rational bases

Full solution

Q. Write an exponential function in the form

y=a b^{x}

that goes through the points

(0,7)

and

(3,3584)

.

\newline

Answer:

Find a Value: We need to find the values of $a$ and $b$ in the exponential function $y=ab^{(x)}$ that satisfy the given points $(0,7)$ and $(3,3584)$ . Using the first point $(0,7)$ , we substitute $x=0$ and $y=7$ into the equation to find the value of $a$ . $y = ab^{(x)}$ $b$ $0$ Since any number raised to the power of $b$ $1$ is $b$ $2$ , we have: $b$ $3$ Therefore, $b$ $4$ .
Solve for b: Now we use the second point $(3,3584)$ and the value of $a$ we just found to solve for $b$ . Substituting $x=3$ , $y=3584$ , and $a=7$ into the equation, we get: $3584 = 7b^{3}$ To find $b$ , we divide both sides by $7$ : $b^{3} = \frac{3584}{7}$ $b^{3} = 512$ To find $b$ , we take the cube root of both sides: $b = 512^{\frac{1}{3}}$ $b = 8$
Final Exponential Function: We now have both values $a$ and $b$ . The exponential function that goes through the points $(0,7)$ and $(3,3584)$ is: $\newline$ $y = 7 \cdot 8^{x}$ $\newline$ This is the final answer.

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