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I'm new here, pretty much a maths beginner but with a huge interest in this art, English not being my first language, and I have a lot of questions that I hope people will not laugh at me for (especially for the terminology).

My very first one is:

I wanted to learn how to calculate an exponential function based on certain points on the curve, and how to determine a value on an exponential curve based on that underlying function. In this case, I wanted a curve where y increases five-fold everytime x increases by $20$. Example: $[x = 20; y = 5], [x = 40; y = 25], [x = 60; y = 125], [x = 80; y = 625], [x = 100; y = 3125]$.

But I neither understand how to put that into a formula, nor how to, i.e., calculate $y$ for, say, an $x$ of $25$ or $30$.

kimchi lover
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Tathy Nobble
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2 Answers2

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You want $y$ to get multiplied by $5$ when $x$ increases by $20$, or $\frac{x}{20}$ increases by $1$. This is can be achieved by the function $$y=5^{\frac{x}{20}}$$ Just input an $x$-value to get the corresponding $y$-value in this.

Vishu
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Take y = $ae^{\alpha x} + c$ as it is an exponential function

y increases by the power of 5 whenever x increases by 20. So,

$\alpha$ is $\frac{1}{20}log_e(5)$. ($log_e5$ was multiplied to change the base from e to 5. If it increased by a power of 10, multiply by $log_e10$, which is roughly 2.303)

or, y = a.$5^{x/20}$ + c

Plug the numbers you gave in this equation, and you will find that a and c are 1 and 0 respectively. Now you can choose any value of x to get y.

Also don't ever feel shy of asking questions. We are here to help :)

cybershiptrooper
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  • I feel stupid. If, i.e., x = 60, then 60/20 = 3 and 3 x 5 = 15, not 125 :( – Tathy Nobble Dec 23 '20 at 20:00
  • It isn't 3x5, but $5^3$ which means 5x5x5. So $5^x$ is 5x5x5...x times(yes, we can still work with decimals in this function- they just become 'irrational'). This is the very essence of exponential functions. If you multiply 5 by x, we are actually calculating 5x (which is called a polynomial function). – cybershiptrooper Dec 23 '20 at 20:11
  • Also $e^{log_e(5)}$ is just 5. And $xlog_e(5)$ is $log_e(5^x)$. These are properties of the logarithmic function(which is the opposite of the exponential function). Try combining the two properties. – cybershiptrooper Dec 23 '20 at 20:14
  • Oh, I gotcha! I mistook the exponentiation for a multiplication. Thanks so much for clearing that up! – Tathy Nobble Dec 23 '20 at 20:18