The given problem involves solving for the constants $a$ and $b$ in the exponential function $f(x) = a b^x$, given two points $f(4) = 8$ and $f(8) = 45$. Then, we calculate $f(5)$ using the obtained equation.

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Step 1: Write the function for the given points.

Using $f(4) = 8$, we have: 8 = a b^4 \tag{1}

Using $f(8) = 45$, we have: 45 = a b^8 \tag{2}

Step 2: Solve for $b$.

Divide equation (2) by equation (1): $\frac{45}{8} = \frac{a b^8}{a b^4}$ $\frac{45}{8} = b^{8-4}$ $\frac{45}{8} = b^4$ Take the fourth root of both sides: $b = \left(\frac{45}{8}\right)^{\frac{1}{4}}$

Numerical calculation: $b = \left(\frac{45}{8}\right)^{\frac{1}{4}} \approx 1.71$

Step 3: Solve for $a$.

Substitute $b \approx 1.71$ into equation (1): $8 = a (1.71)^4$ Calculate $(1.71)^4$: $1.71^4 \approx 8.599$ $a = \frac{8}{8.599} \approx 0.93$

Step 4: Find $f(5)$.

The function is now approximately: $f(x) = 0.93 (1.71)^x$

Substitute $x = 5$: $f(5) = 0.93 (1.71)^5$ Calculate $(1.71)^5$: $1.71^5 \approx 14.706$ $f(5) = 0.93 \times 14.706 \approx 13.68$

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Final Answer:

$f(5) \approx 13.68$

Would you like a breakdown of any specific step?

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Related Questions:

1. How do you derive the formula for an exponential function given two points?
2. What are some real-world applications of exponential functions?
3. How can you solve exponential equations with logarithms?
4. What is the relationship between the base $b$ and the growth rate in an exponential function?
5. How can exponential models be used to fit data points?

Tip:

When solving exponential functions, logarithms are a powerful tool to isolate variables and verify solutions efficiently.