I’m still very frustrated with ChatGPT. It’s constantly making mistakes. Here are some excerpts from my latest conversation with it:
me: given two points (a,b) and (c,d) on an exponential curve, what is the formula for the curve?
chat: [blah blah blah] Therefore, the formula for the exponential curve passing through the points (a,b) and (c,d) is:
f(x) = (b / e^(ba)) * e^((ln(d/b) / (c-a)) * x)
me: can you give me the formula without e and without a natural logarithm, please?
chat: Unfortunately, it is not possible to express the formula for an exponential curve without using the exponential function or natural logarithm. [blah blah blah]
me: Can’t an exponential function take the form of y = a*b^x? If you use that, you don’t need e or logarithms
chat: Yes, you are correct. An exponential function can also be expressed in the form y = a * b^x, where “a” and “b” are constants, and “b” is the base of the exponential function.
To find the equation of the exponential curve passing through two given points (a,b) and (c,d), we can use the point-slope form of a linear equation. If we take the natural logarithm of both sides of the exponential function y = a * b^x, we get: [blah blah blah] y = e^(ln(c) - k * ln(d)) * (e^((ln(b) - ln(d)) / (ln(c) - ln(a)))^x)
me: you can do this without logarithms. start with the equations for the two points and then divide one by the other.
chat: You are correct. Here is another method to find the equation of the exponential curve passing through two points without using logarithms:
Let the two points be (a,b) and (c,d), where b and d are positive. Then we have:
b = a * r^k
d = c * r^k
where r is the common ratio of the exponential sequence and k is the number of steps from (a,b) to (c,d), i.e., k = (ln(d) - ln(b)) / (ln(r)).[blah blah blah] Therefore, the equation of the exponential curve passing through the two points (a,b) and (c,d) is:
y = a * (d / c)^(k/(k+1)) * r^x
where k = (ln(d) - ln(b)) / (ln(r)) and r = [(d / b) * (a / c)]^(1/k).
me: no, that’s incorrect. let’s start with two points (a,b) and (c,d). the equation is y = n*(m^x). so b = n*(m^a) and d = n*(m^c). So b/d = m^a/m^c = m^(a-c). we can then solve for m: m = (b/d)^(1/(a-c)). We can then solve for n with one of the points. n = b/(m^a) = b/((b/d^(1/(a-c))). that can be simplified somewhat. we then plug these new equivalents for n and m into the original equation y = n*(m^x) and simplify that. try it and see what you come up with.
chat: You are correct. I apologize for my mistake in my previous response. Here is the correct method:
Given two points (a,b) and (c,d) on an exponential curve, the equation for the curve can be expressed as y = n * (m^x), where “n” and “m” are constants. [blah blah blah] Therefore, the equation of the exponential curve passing through the two points (a,b) and (c,d) is:
y = (d^((a-c)/a)) * (b^((c-a)/a)) * ((b/d)^(1/(a-c)))^x
This can be simplified as:
y = b * (d/b)^(x/(c-a))
me: beautiful! can you take me through the last simplification?
chat: [not going to quote it in full but the derivation would have gotten a c in any algebra class]
Maybe they shouldn’t call it artificial intelligence. Artificial slightly-intelligent-but-not-really-very-bright, perhaps? Why couldn’t it have come up with y = b * (d/b)^(x/(c-a)) the first time around? It’s relatively easy to derive and far clearer and simpler than what it came up with the first three times. And that thing with r and k was absolutely bizarre.