Questions: Find a power function through the points (1,7) and (3,49). f(x)=7 x log(7)/log(3)

Solution Steps

To find a power function of the form \( f(x) = ax^b \) that passes through the points \((1,7)\) and \((3,49)\), we need to determine the constants \(a\) and \(b\). We can set up a system of equations using these points and solve for \(a\) and \(b\).

1. Substitute the point \((1,7)\) into the equation to get \(7 = a \cdot 1^b\), which simplifies to \(a = 7\).
2. Substitute the point \((3,49)\) into the equation to get \(49 = 7 \cdot 3^b\).
3. Solve for \(b\) using the equation \(49 = 7 \cdot 3^b\).

Using the point \((1, 7)\), we substitute into the power function \( f(x) = ax^b \): \[ 7 = a \cdot 1^b \implies a = 7 \]

Next, we use the point \((3, 49)\) to find \( b \): \[ 49 = 7 \cdot 3^b \] Dividing both sides by 7 gives: \[ \frac{49}{7} = 3^b \implies 7 = 3^b \] Taking the logarithm of both sides: \[ \log(7) = b \cdot \log(3) \implies b = \frac{\log(7)}{\log(3)} \approx 1.7712 \]

Now that we have both \( a \) and \( b \), we can express the power function: \[ f(x) = 7 \cdot x^{1.7712} \]

To ensure the function is correct, we can check the values at the given points:

• For \( x = 1 \): \[ f(1) = 7 \cdot 1^{1.7712} = 7 \]
• For \( x = 3 \): \[ f(3) = 7 \cdot 3^{1.7712} \approx 49 \]

Final Answer