Question

From the problem I obtained the equation "1800(2^(x/40))." I need to convert that to an equation that looks like this

A(e^kx).

What do A and k need to be for that to be true?

No idea how to do this. Can someone help?

Answer 1

\[\log_b x^n = n \log_b x\]I think that should help you.

Answer 2

I'm honestly still lost. What happens when I take the ln of 1800(2^x/40)?

Answer 3

Does it become x/40 (ln(1800(2)) ?

Answer 4

I misread it. It's not that complicated. Sorry.

Answer 5

Okay lol. I still don't completely understand the whole e thing and how it relates to exponential growth

Answer 6

Essentially what you have is \(2^{\frac{1}{40}x} = e^{kx}\)
You can just say A = 1800 and stick that back in later.

Answer 7

Okay. So how do I get k then?

Answer 8

Thanks so much for the help btw.

Answer 9

I think you just have to think of it as
\[e^{ln 2^{\frac{1}{40}}x}\]

Answer 10

That 1/40 is the exponent of the 2, by the way. It's hard to tell in that tiny font.

Answer 11

Okay. I see. So my final answer would be

1800(e^xln2^1/40)

Answer 12

I am not 100% sure about this! But I think that's how you convert bases like that.

Like any number can be written as \(n = e^{\ln n}\)

Answer 13

Okay. I do have 3 guesses on my online assignment, so if it ends up not being right I'll let you know.

1 sec

Answer 14

So A was right, it is 1800, but k was not.


So yeah, we are left with how to convert 2^(x/40) to e^kx

Answer 15

Ok, hold on. I think I get it now.

Answer 16

\[2^{\frac{1}{40}x} = e^{kx}\]\[\frac{1}{40}x = \log_2 e^{kx}\]\[\frac{1}{40}x = \frac{\ln e^{kx}}{\ln 2}\]\[\frac{1}{40}x =\frac{kx}{\ln 2}\]\[\frac{\ln 2}{40}x = kx\]\[\frac{\ln 2}{40} = k\]

Answer 17

Tell me if that makes sense. :D

Answer 18

That makes a lot of sense actually.

Answer 19

I think the first time I tried to shortcut doing all that math and bungled something somewhere. But that one should work. And I can actually even explain it now, if needed. :P

Answer 20

It was the wrong answer, even though I'm pretty sure it's right :/

Ugh. Okay, it's fine, I'm pretty sure I understand how you did that. I appreciate the help.

Answer 21

Ok, well, I don't know what they're expecting, then. :(
Sorry.

Answer 22

I have 1 guess left, but I'm not sure how I should enter it. Like, I'm fairly certain the answer you gave is right, cause I actually understand how you got it and it makes sense to me

Answer 23

Also, I just noticed, I had it right before, too, just written in a weird form. But they are equal because of the very first thing I said. :D

Answer 24

Ah yeah I see.

Answer 25

\[\frac{1}{40} \times \ln 2 = \ln 2^{\frac{1}{40}}\]
Now I'm double convinced!

Answer 26

Alright. I'll try to figure out how they want it entered, I really appreciate the help dude

Answer 27

I understand it a lot better now, thanks again.