Math  /  Algebra

QuestionFind a possible formula for the exponential function described. f(6)=63,f(65)=18f(6)=63, \quad f(65)=18
Round your answers to four decimal places. f(x)=abxf(x)=a b^{x}, where: a=a= i b=b= i

Studdy Solution

STEP 1

1. The function f(x)=abx f(x) = a b^x is an exponential function.
2. We are given two points on the function: f(6)=63 f(6) = 63 and f(65)=18 f(65) = 18 .
3. We need to find the values of a a and b b and round them to four decimal places.

STEP 2

1. Substitute the given points into the exponential function.
2. Solve the system of equations to find a a and b b .
3. Round the solutions to four decimal places.

STEP 3

Substitute the given points into the exponential function.
For f(6)=63 f(6) = 63 : 63=ab6 63 = a b^6
For f(65)=18 f(65) = 18 : 18=ab65 18 = a b^{65}

STEP 4

Solve the system of equations to find a a and b b .
First, divide the second equation by the first to eliminate a a :
1863=ab65ab6 \frac{18}{63} = \frac{a b^{65}}{a b^6}
This simplifies to:
1863=b656 \frac{18}{63} = b^{65-6}
1863=b59 \frac{18}{63} = b^{59}

STEP 5

Solve for b b by taking the 59th root of both sides:
b=(1863)159 b = \left( \frac{18}{63} \right)^{\frac{1}{59}}
Calculate b b and round to four decimal places:
b(1863)1590.9874 b \approx \left( \frac{18}{63} \right)^{\frac{1}{59}} \approx 0.9874

STEP 6

Substitute b0.9874 b \approx 0.9874 back into one of the original equations to solve for a a .
Using 63=a(0.9874)6 63 = a (0.9874)^6 :
a=63(0.9874)6 a = \frac{63}{(0.9874)^6}
Calculate a a and round to four decimal places:
a63(0.9874)667.2676 a \approx \frac{63}{(0.9874)^6} \approx 67.2676

STEP 7

The possible formula for the exponential function is:
f(x)=67.2676×0.9874x f(x) = 67.2676 \times 0.9874^x

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