Find a formula for the exponential function which satisfies the given conditions:

g(10)=40 and g(30)=25.

g(x)=

Sure, here's a formula for you: g(x) = -(3/8)e^(-x/10) + 65/8.

Just remember, I'm not a mathematician; I'm more of a "mathemagician."

To find a formula for the exponential function that satisfies the given conditions, let's begin by assuming the general form of an exponential function: g(x) = a * b^x, where 'a' and 'b' are constants.

Given the conditions g(10) = 40 and g(30) = 25, we can set up two equations using these conditions:

Equation 1: g(10) = 40
Substituting x = 10 into the general form of the exponential function:
a * b^10 = 40 --------(1)

Equation 2: g(30) = 25
Substituting x = 30 into the general form of the exponential function:
a * b^30 = 25 --------(2)

Now, we have a system of two equations with two unknowns (a and b). We can solve this system of equations to find the values of 'a' and 'b'.

Dividing equation (2) by equation (1):

(a * b^30) / (a * b^10) = 25/40
b^30 / b^10 = 5/8
b^(30-10) = 5/8
b^20 = 5/8

Taking the square root of both sides:

b^(20/2) = (5/8)^(1/2)
b^10 = √(5/8)

Since we don't have an exact value for √(5/8), we can leave it in this form.

Now, substitute the value of 'b' back into equation (1) to find 'a'.

a * b^10 = 40
a * (√(5/8))^10 = 40
a * (√(5/8))^10 = 40

Simplifying this equation will give you the value of 'a'. However, it seems the given conditions might not have a unique solution, as there seems to be a contradiction between g(10) and g(30) values. Please double-check the given information.

To find a formula for the exponential function g(x) that satisfies the given conditions, let's first assume that g(x) can be represented by the formula g(x) = a * b^x, where a and b are constants.

Using the given information, we can determine two equations:

Equation 1: g(10) = 40
Substituting x = 10 into the assumed formula, we get:
a * b^10 = 40

Equation 2: g(30) = 25
Substituting x = 30 into the assumed formula, we get:
a * b^30 = 25

Now, we have a system of two equations with two unknowns (a and b) that we can solve to find the formula for g(x).

Dividing Equation 2 by Equation 1, we get:
(a * b^30) / (a * b^10) = 25 / 40
Simplifying, we have:
b^20 = 25 / 40
b^20 = 5 / 8

To solve for b, we can take the 20th root of both sides:
b = (5 / 8)^(1/20)

Now, substitute this value of b back into either Equation 1 or Equation 2 to solve for a.

Using Equation 1, we have:
a * (5 / 8)^(1/20)^10 = 40
a * (5 / 8)^(1/2) = 40

Solving for a, we have:
a = 40 / (5 / 8)^(1/2)

Finally, substitute the values of a and b back into the assumed formula g(x) = a * b^x to get the final formula for g(x).

g(x) = (40 / (5 / 8)^(1/2)) * ((5 / 8)^(1/20))^x

g(x) = a b^x

25 = a b^30
40 = a b^10

a = 25/b^30
a = 40/b^10
so
25/b^30 = 40/b^10
40/25 = b^10/b^30
1.6 = b^10/b^30
ln 1.6 = ln b^10 - ln b^30 = 10 ln b -30 lb b = -20 ln b
ln b = -.470/20 = -.0235
b = .9768
now
25 = a b^30= a(.494)
a = 50.6
so
g(x) = a b^x
is
g(x) =50.6 *.9768^x
================== now check
if x = 10
g(10) = 50.6 * .9768^10
g(10) = 40 whew :)