Questions: An exponential function f(x)=ab^x passes through the points (0,7000) and (3,7). What are the values of a and b ? a= b=

Solution Steps

Using the point \( (0, 7000) \), we can find the value of \( a \) in the exponential function \( f(x) = a b^x \). Since \( b^0 = 1 \), we have:

\[ f(0) = a \cdot b^0 = a = 7000 \]

Thus, we find:

\[ a = 7000 \]

Next, we use the second point \( (3, 7) \) to find the value of \( b \). Substituting into the function gives us:

\[ f(3) = a \cdot b^3 = 7 \]

Substituting \( a = 7000 \) into the equation, we have:

\[ 7000 \cdot b^3 = 7 \]

To isolate \( b^3 \), we divide both sides by 7000:

\[ b^3 = \frac{7}{7000} \]

This simplifies to:

\[ b^3 = \frac{1}{1000} \]

Taking the cube root of both sides, we find:

\[ b = \sqrt[3]{\frac{1}{1000}} = \frac{1}{10} \]

Final Answer