Mike_bb's latest activity
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MMike_bb reacted to bhobba's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with
Like.
It is its definition. You really need to see how all this stuff follows from the definition of ln(x) We define ln(x), x > 0, called... -
MMike_bb reacted to mathwonk's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with Like.
It helps to have a definition of e. One definition is that e is the unique base such that the derivative of e^y at y=0 equals 1. Since...
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MMike_bb replied to the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##?.Maybe it's elegant proof, but I like your proof because it shows where ##e^y## came from.
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MMike_bb reacted to Sagittarius A-Star's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with Like.
Here, in the spoiler, I found the same proof, written in a more elegant way: $$\int\frac1x\,dx,\quad...
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MMike_bb reacted to Sagittarius A-Star's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with Like.
you're welcome
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MMike_bb replied to the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##?.Sagittarius A-Star Big thanks!!! I appreciate you very much! Now I understand how it works!
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MMike_bb reacted to Sagittarius A-Star's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with
Informative.
You may differentiate the function ##F(y)=\int_1^{e^y}\frac{1}{t}\,dt## without using the ##ln(x)## function, simplify ##F'(y)## and... -
MMike_bb reacted to Sagittarius A-Star's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with Informative.
According to the chain rule: ##F'(y)=\frac{1}{e^y}\cdot {d \over dy}(e^y)## Maybe you continue the calculation first for ##e## and then...
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MMike_bb reacted to Sagittarius A-Star's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with Informative.
Then you would not get ##F(y) = y## after integrating.
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MMike_bb reacted to Sagittarius A-Star's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with Like.
That's not true. Proof: https://www.wolframalpha.com/input?i2d=true&i=Integrate%5BDivide%5B1%2Ct%5D%2C%7Bt%2C1%2CPower%5B10%2Cy%5D%7D%5D
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MMike_bb reacted to Sagittarius A-Star's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with Informative.
The simplification in posting #6 goes: ##F'(y)=\frac{1}{e^y}\cdot {d \over dy}(e^y) = 1##.
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MMike_bb reacted to Sagittarius A-Star's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with
Agree.
In Wolfram Alpha they write ##log()## for the natural logarithm, which is usually written as ##ln()##. They mention it: -
MMike_bb reacted to Sagittarius A-Star's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with Like.
Please see above in posting #13.
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MMike_bb reacted to Sagittarius A-Star's post in the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##? with Like.
If you put ##10^y## as upper limit, then you would get ##F'(y)=\frac{1}{10^y}\cdot {d \over dy}(10^y) \neq 1## and ##F(y) \neq y##.
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MMike_bb replied to the thread Undergrad Why is ##x=e^y## the inverse of ##y=\int_1^x \frac{1}{t} dt##?.Sagittarius A-Star In your proof you use the fact that ##exp(x)'=exp(x)##, right?