Question 6

An exponential function $f(x)=a b^{x}$ passes through the points $(0,8000)$ and $(2,320)$. What are the values of $a$ and $b$ ?
\[
\begin{array}{l}
a=\square \\
b=\square
\end{array}
\]
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Step 1 ：First, let's substitute the point (0,8000) into the function $f(x)=a b^{x}$ to get the value of $a$. We get the equation $a \cdot b^{0} = 8000$. Since any number raised to the power of 0 is 1, we can simplify this equation to $a = 8000$.

Step 2 ：Next, we substitute the point (2,320) into the function to get an equation in terms of $b$. We get the equation $8000 \cdot b^{2} = 320$.

Step 3 ：Finally, we solve this equation to get the value of $b$. Dividing both sides of the equation by 8000, we get $b^{2} = \frac{320}{8000} = \frac{1}{25}$. Taking the square root of both sides, we get $b = -\frac{1}{5}$ or $b = \frac{1}{5}$. Since the function is decreasing, we take $b = -\frac{1}{5}$.

Step 4 ：So, the values of $a$ and $b$ are $a = \boxed{8000}$ and $b = \boxed{-\frac{1}{5}}$.