Exponential Function Puns

I've been thinking about this for a while. I understand that if: y = a ^ x then the inverse is x = log_a (y). But what I want to know is what that relationship represents. If I were to graph both functions, how can I interpret the values of the logarithmic function (as I understand, the exponential function represents exponential growth)? And how the values returned by the function in the same value X are related to each other? Hope this makes sense. Thanks in advance!

Hello community,

I'm trying to solve some applied task. The task is to approximate experimentally obtained curve to express required parameters. The curve can be approximated via function f(t)=a/(a+b)*R*(1-exp(-(a+b)*t)), where t - time, R & b - known coefficients = const, a - required parameter. I apply least squares method to pick parameter a and everything works well, experimental and approximated curves correlate. BUT! and here is a problem: at some value of a the curve is no more sensitive to variation. I mean it changes when vary from 0.1 to 1, but change is negligible when vary from 1 to ∞ . It is expected since for a -> ∞ , limit f(t) = R. QUESTION: How to make the function more sensitive to variation of a in a range 1 < a < ∞. I would appreciate any suggestions and recommendations. Thank you!

Additional dispensable info: the equation you see above is "integrated eq. of association", solved PDE of Langmuir adsorption. a is a product of multiplication of concentration and coefficient of association rate. I've tried to transform f(t) into Taylor series up to 6th order but it is not so helpful as expected and associated with additional problems...

Hi, I'm studying for a maths exam and I have a question which is:

q(t) = Qe^-(t/(R*C))

where Q is the initial charge (measured in Coulombs), and t is time (measured in seconds). Suppose Q = 70 and R = 30.

The question asked is:

Suppose that for a different capacitor we are given that 30% of the initial charge is lost in the first 8 seconds. Find the value of C.

The Solution is : 4/(15*ln(10/7) = 0.7476

Now I believe that the way to solve this question is to rearrange the exponential function, though I have tried this and cannot manage to get an equation similar to the solutions. If anyone could lend me a hand I would be very thankful.

import numpy as np

# parameters calculated with partial_fractions.py
numerators = np.array([(-0.06867707087211337-6.017389477372975e-17j), (-0.05958278929352701+0.07121671118312572j), (-0.05958278929352722-0.07121671118312578j), (0.16882680902709102+0.0966123090073435j), (0.16882680902709107-0.09661230900734338j), (-0.007559832563919337-0.14490420850010693j), (-0.007559832563919344+0.14490420850010693j), (-0.06734565173358785-0.02984056574559885j), (-0.06734565173358782+0.029840565745598843j)])
zeros = np.array([(0.9165697260294747+0j), (0.6737558584892636+0.6810867407245075j), (0.6737558584892636-0.6810867407245075j), (0.12493819008001533+1.1354149216147027j), (0.12493819008001533-1.1354149216147027j), (-0.402976045293055+0.9488264695079811j), (-0.402976045293055-0.9488264695079811j), (-0.85400286629096+0.3577698511391375j), (-0.85400286629096-0.3577698511391375j)])


# calculates number of fish with lifetime 0 at step n if at the beginning there was only one fish with lifetime 6
# the explicit expression of the function is a sum of exponential
def func(n):
    return round(np.sum(-numerators * zeros ** (-n-1)).real)


# deduce the total number of fish from the number of fish with lifetime 0 in the last 9 steps
def fish(steps, x):
    c = [1, 1, 2, 2, 2, 2, 2, 2, 1]
    return sum(np.multiply(list(map(lambda n: func(n + 6 - x), range(steps - 8, steps + 1))), c))


with open('input', 'r') as file:
    numbers = list(map(int, file.read().split(',')))
print(sum(map(lambda x: fish(80, x), numbers)))
print(sum(map(lambda x: fish(256, x), numbers)))

I found the closed form of the number of fish after n steps. The process is similar to the one to find the closed form of the Fibonacci function. The resemblance is no surprise because Fibonacci was inspired by animals reproduction.

This program solves the puzzle with an exponential function you can write down on a piece of paper. For explanation on the process to find the function check this file.

The constant parameters are calculated with partial_fractions.py.

Hey all,

A computer game that I play applies a scaling function to income generation that means each additional resource gives less of a return. I'm trying to work out what the function is based on some values I've observed. It appears to be exponential based on plotting the points, but I don't think it's as simple as Y = AX^B because solving that for different pairs of points gives different values. I wondered if anyone could suggest how I might tackle this?

The values that I've observed are as follows:

Where X is the amount of resource and XY essentially gives the income per turn.

p.s. There is also a small random factor involved, but I'm pretty sure it's small enough to not majorly affect the calculation.

https://ibb.co/DQb3DYB

The function goes through the "nice" points (0,-2), (-1,-5). The answer must be in the form f(x)=ab^x

Plugging in and solving,

-2 = ab^0 (b^0 = 1) a = -2

-5 = (-2)b^-1 -5/-2 = b^-1

5/2 = 1/b

5/2b = 1

b = 1/(5/2)

b= 0.4

QED f(x)=(-2)(0.4)^x

Now this is obviously wrong as -2 would make this a declining function, which visibly it is not, but I don't understand how I could've gotten A wrong when it's literally the y-intercept..

Is the exponential function y=5^x the same as Y=5^x? Is the capital Y different from y? Thank you in advance gents!

Their last big fight:

e^x: “Pi, I can never figure you out!”

Pi: “Me? The more you seem to change, the more you just stay the same!”

https://imgur.com/a/mSE3Uks

https://imgur.com/a/h9VmiiH

Any help appreciated especially with that last one

I am trying to find the official name of this problem, but my google searches are turning up nothing. I ran across it while programming. I appreciate any help you can provide.

An exponential equation with the relevant variables is as follows.

https://preview.redd.it/vsermjl6i1681.png?width=150&format=png&auto=webp&s=0dfa762ad06288fcd557f5e8c2bd6f920d0602ca

I know that a is the sharpness of growth, b is the asymptote, and c translates the graph along the x-axis.

Different exponential functions, with the same rate of growth, all intercepting the y-axis at 1 can be drawn as follows.

https://preview.redd.it/fu81ofdci1681.png?width=744&format=png&auto=webp&s=3807d97afca5b31677acd45c919586f5d862806c

We can change c to g(b) given that the y-intercept is constant when b and c change, so I assume b and c are related.

https://preview.redd.it/dga9chvoi1681.png?width=189&format=png&auto=webp&s=f757048de3ebd6398ffa5792645167b7001ed8b6

If we add the intercept itself as a variable, the problem might be expressed as follows, where we use c as our y-intercept.

https://preview.redd.it/lmuw6nn6j1681.png?width=175&format=png&auto=webp&s=c6b9bbe88746b9e9285e3c516e158e80ac2d3bdc

We can see that g(b,2) is a different set of irrational numbers than g(b,1).

https://preview.redd.it/ksy6dj09m1681.png?width=744&format=png&auto=webp&s=d8a8c3af6be77d4b3a026f4ef7edaec8242287a5

I tried plotting g(b,c) but I'm not good enough at math to figure out what kind of function it is.

What is this problem called? Can you show me the mathematician who solved it, or direct me to the third grade algebra lesson that I was zoned out during? Thanks.

You deposit money in an account earning 5% interest compounded monthly. How long will it take your investment to double?

So I got (1+.05/12)^12t = 2 since the original amount is x, and the end amount is 2x so dividing by x cancels it out. However, when I enter the log of this I get bizarre answers.

Also having problems with this:

A bacteria culture initially contains 2000 bacteria and doubles every half hour.

Find the size of the baterial population after 80 minutes.

Find the size of the baterial population after 4 hours.

Let's says that we are modelling the growth of population of bacteria

**No of bacteria after x seconds is given by **

P(x) =5e^x (5 is the initial no of bacteria ,at the beginning )

After 1second (or) when 1 second runs in the clock we will have 5*e ~13.59 (or) thirteen full bacteria and a small chunk .

Here what do mean by continuous growth ,we say that when the clock runs from 0 to 1 second it grows in every instant ,but what I don't get is that the it grows every instant ,but at what rate ,the growth rate for the whole second is 100% .

https://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

Pls use this for reference.

And what is the meaning behind the growth constant ,what does it mean to say, that a population of bacteria grows P(x) =5e^(kx).

What is the physical meaning of the constant?

If 2000 dollars is invested in a bank account at an interest rate of 10 per cent per year,

Find the amount in the bank after 14 years if interest is compounded monthly:

I did 2000(1+.0083)^168 and I got 8018.8 and it says its incorrect.

For continuously:

2000^.1(14) and I get 41825.58 and it also says it's incorrect.

Is it an error?

Write the exponential function f(x) = ca^(rx). in that form f(x) =de^(bx) . Fin d and b

how do I go about solving these kinds of problem

MOST IMPORTANT EDIT: if your high school didn't require calculus to graduate and/or required stats to graduate, I am just being picky and you probably have little reason to read further.

As an engineer, I understand calculus is a very important requisite in most STEM disciplines and I think it should still be taught in high school. However, calculus just doesn't apply to many issues we face in everyday adult life (okay, that part is probably a popular opinion but I think my solution is unpopular).

I think the concepts behind exponential functions ( e.g. 2^x) are very important for everyone to understand however because that's essentially the gist of compound interest and personal finances.

Similarly, I believe some understanding of statistics is very important for everyone because it lets you interpret data in a more "statistically significant" manner. And let me tell you, not only are you constantly surrounded by data... you are data.

TLDR We apply exponential functions and statistics in everyday life more often than we apply n-degree polynomial calculus, so schools should focus on those more.

Edit: Apparently there is a lot more variety in how high schools teach math than i thought. My high school had precalc AND calculus as mandatory classes. We offered AP stats but only ~20/300 kids in my class took it.

Edit2: Also some comments helped me realize i specifically meant n-degree polynomial functions are less applicable than exponential functions... in my school our calculus classes were almost exclusively taught using polynomial functions. Im not arguing exponential functions deserve their own year long course but that they deserve more love in high school curriculums.

Hi, I’m very new to QM. I’m working on a practice problem in which I have to solve the 1D particle in a box problem without using trig functions in my wave function and instead use complex exponentials. I chose the general form Ae^ikx + Be^-ikx but when you set x equal to the boundary conditions you can no longer solve the problem unless I’m missing something. At x=0, A=-B since psi(0)=0. At x=a, psi(a)=0 and Ae^ikx=-Be^-ikx so e^ikx=e^-ikx. The only time that’s true if I’m not mistaken is when x=0, which ruins our solution. At this point when using trig functions, we can introduce n by saying sin(ka)=0, so ka=n*pi, which is really handy. How can I solve this problem without that? Thanks for any help. (And no, this is not an exam or homework problem)

The first cell calculates e^-x with its series.

The second calculates e^x and after that, it computes 1/e^x = e^-x

Why am I getting the wrong result on the first method??
https://imgur.com/a/oz4L5VT

Hi there! Its been far too long since i've had my math classes, so i could really use some help on this one. I'll briefly explain what i'm trying to apply the math for, then explain what I think the math looks like.

In the D&D game I'm setting up, travelling overseas is something considered very dangerous. Merchants will of course still rely on the usage of boats as they can transport things in far bigger bulk than a horse cart could. For the sake of simplicity, there are 2 types of boat: a carrying vessel for trading goods, and an armed vessel for protection. I want to have a simple function to calculate the chance of a carrying vessel arriving at it's destination when paired with an x amount of armed vessels.

For this function, I'd appreciate the numbers to correspond to something easily usable with dice. my first thought is that a carrying vessel with 0 armed vessels has a 33% chance of arriving (failure corresponding to a 1 or a 2 on a 6-sided dice), and a 95% chance of arriving with 4 vessels (failure corresponding to a 1 on a 20-sided dice). Adding any more vessels than this should add very little reward, capping the probability at 99%, so that the chance of failure will always persist.

so with y being the chance at success and x being the amount of armed vessels, what would my function end up looking like?