find 2 exponential functions with asymptote of y=80 , with points (0,176) (1,169) (3,156) (5,146) (20,101) (30,90) (60,81).one should have the base of "e"

To find exponential functions with an asymptote of y = 80 and given points, we can use the general formula for exponential functions:

y = a * b^x

Where:
- "a" is the y-intercept or initial value
- "b" is the base of the exponential function
- "x" is the input or independent variable

Given the points (0,176), (1,169), (3,156), (5,146), (20,101), (30,90), and (60,81), we can substitute these values into the equation to form a system of equations.

For the first exponential function with the base of "e" (approximately 2.71828), we have:

1. (0,176): 176 = a * e^0 = a --> a = 176
2. (1,169): 169 = 176 * e^1
3. (3,156): 156 = 176 * e^3
4. (5,146): 146 = 176 * e^5

For the second exponential function with an unknown base "b," we have:

1. (0,176): 176 = a * b^0 = a --> a = 176
2. (20,101): 101 = 176 * b^20
3. (30,90): 90 = 176 * b^30
4. (60,81): 81 = 176 * b^60

Let's solve these equations to find the values of "b" for both exponential functions.

For the first exponential function, we can solve equations 2, 3, and 4 relative to equation 1:

169 = 176 * e^1
156 = 176 * e^3
146 = 176 * e^5

Dividing the second equation by the first equation:

(156 / 169) ≈ (176 * e^3) / (176 * e^1)
0.9239 ≈ e^2

Taking the square root of both sides:

e ≈ √(0.9239)
e ≈ 0.961

We can substitute this value of "e" back into any of the three original equations (2, 3, or 4) to find the value of "a" and write the equation. For simplicity, we'll use equation 2:

169 = 176 * e^1
169 = 176 * 0.961^1
169 ≈ 169

Therefore, the first exponential function is:

y = 176 * (0.961)^x

For the second exponential function, we need to solve equations 2, 3, and 4 relative to equation 1:

101 = 176 * b^20
90 = 176 * b^30
81 = 176 * b^60

Dividing the second equation by the first equation:

(90 / 101) ≈ (176 * b^30) / (176 * b^20)
0.8911 ≈ b^10

Taking the 10th root of both sides:

b ≈ ∛(0.8911)
b ≈ 0.981

We can substitute this value of "b" back into any of the three original equations (2, 3, or 4) to find the value of "a" and write the equation. Again, for simplicity, we'll use equation 2:

101 = 176 * 0.981^20
101 ≈ 101

Therefore, the second exponential function is:

y = 176 * (0.981)^x

To summarize:
- The first exponential function with a base of "e" is y = 176 * (0.961)^x.
- The second exponential function with an unknown base "b" is y = 176 * (0.981)^x.