```python
from IPython.display import Image
from IPython.core.display import HTML 
Image(url= "https://i.imgur.com/WBsZZjO.png") # power rule of exponents + an estimation
```









```python
Image(url= "https://i.imgur.com/OKzEasd.png")
```









```python
from sympy import *
from sympy import euler
x = symbols('x')
c = symbols('c', real = True)
sympify(c).is_real
```




    True




```python
eq5 = c*E**(x)
Eq(eq5,100) #solving for c when x = 10 because the plot looks nice
```




$\displaystyle c e^{x} = 100$




```python
plot(E**(x-5.4)) # after plotting this I noticed that it would be nice if this y= 100 right when x = 10
                 # the final solution may be useful for conceptualizing percentage of exponential growths
```


```python
Image(url= "https://i.imgur.com/F2uZGSM.png")
```









```python
eq11 = Eq(c*E**(10),100) 
eq11
# This is what I want to solve. 
```




$\displaystyle c e^{10} = 100$




```python
eq1 = 0.00453999208646404*exp(x)
Eq(eq1,eq1.subs(x,10).evalf())
```




$\displaystyle 0.00453999208646404 e^{x} = 99.9999804011934$




```python
eq2 = 0.00454*exp(x)
Eq(eq2,eq2.subs(x,10).evalf())
```




$\displaystyle 0.00454 e^{x} = 100.000154708422$




```python
p0 = plot(eq1,eq2,show = False,xlim = (0,10),size = (12,5),legend = True,autoscale = True)
p0[0].line_color = 'r'
p0[1].line_color = 'b'
p0.show()
```


```python
s = solve(eq11)[0]
s # one possible solution for real number c when x = 10
```




$\displaystyle \frac{100}{e^{10}}$




```python
s.evalf()
```




$\displaystyle 0.00453999297624849$




```python
final_solution = (s*E**x)
phi = symbols('phi')
Eq(phi,final_solution)
```




$\displaystyle \phi = \frac{100 e^{x}}{e^{10}}$




```python
final_solution.subs(x,10)
```




$\displaystyle 100$




```python
final_equation = s.evalf()*E**x
Eq(final_equation,100)
```




$\displaystyle 0.00453999297624849 e^{x} = 100$




```python
(0.00453999297624849*E**10).evalf()
```




$\displaystyle 100.0$




```python
p0 = plot(eq1,eq2,final_solution,show = False,xlim = (0,10),size = (12,5),legend = True,autoscale = True)
p0[0].line_color = 'r'
p0[1].line_color = 'b'
p0[2].line_color = 'g'
p0.show()
```


```python
eq14 = Eq(0.00453999297624849*E**(x),100) 
solve(eq14)
```




    [10.0000000000000]




```python
0.00453999208646404 < 0.00453999297624849 < 0.00454
```




    True



37719e8584ca82a67be3eda574cbda77fe93add4
ipynb
Jupyter Notebook
Personal_Projects/Exponential_Growth_y=100_when_x=10_.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Personal_Projects/Exponential_Growth_y=100_when_x=10_.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Personal_Projects/Exponential_Growth_y=100_when_x=10_.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Qwen/Qwen-72B
1. YES
2. YES