#1
|
|||
|
|||
Does anyone have the math skills to isolate the variable, a.
c is an arbitrary constant, a is a real number.
ln(a) ------- = c (1 - a) Isolate a. |
#2
|
|||
|
|||
Re: Does anyone have the math skills to isolate the variable, a.
[ QUOTE ]
c is an arbitrary constant, a is a real number. ln(a) ------- = c (1 - a) Isolate a. [/ QUOTE ] The only equivalence I can find with only one occurence of a is c = \int_1^a (x+1)/x dx where \int_1^a is the definite integral from 1 to a. This is derived from the original equation by first multplying bth sides by (1-a), then exponentiating both sides to get a e^a = e^(c+1). Then taking the natural log of both sides you get c+1 = ln(a e^a) = a + ln(a). Since ln(a) = \int_1^a 1/x dx and a = \int_1^a 1 dx + 1, we have c+1 = \int_1^a (1 + 1/x) dx + 1. But I think this is all you can do. Also, from the second FTC, since c is constant, 0 = (a+1)/a for which the only solution is a = -1, for which the original problem statement is undefined. Anyhow, where does this problem come from? |
#3
|
|||
|
|||
Re: Does anyone have the math skills to isolate the variable, a.
Answer: <font color="white"> Differentiate both sides, leading to 1 / (a * (1-a)) + ln a / (1 - a)^2 = 0. Now substitute in our relationship from before and manipulate to get a = - 1 / c. </font>
EDIT: I don't think this makes a lick of sense. At the very least the result I got doesn't make sense. It's time to go to bed. |
#4
|
|||
|
|||
Re: Does anyone have the math skills to isolate the variable, a.
[ QUOTE ]
fishsauce -- multplying bth sides by (1-a), then exponentiating both sides to get a e^a = e^(c+1). [/ QUOTE ] You have an error here. You turned c(1-a) into c+1-a. PairTheBoard |
#5
|
|||
|
|||
Re: Does anyone have the math skills to isolate the variable, a.
[ QUOTE ]
Answer: Differentiate both sides, leading to 1 / (a * (1-a)) + ln a / (1 - a)^2 = 0. Now substitute in our relationship from before and manipulate to get a = - 1 / c. <font color="white">. </font> EDIT: I don't think this makes a lick of sense. At the very least the result I got doesn't make sense. It's time to go to bed. [/ QUOTE ] Taking the derivative makes no sense. On the left hand side you have a function in the variable a which is Not constant. Just because you're trying to solve for when the function equals c does not mean the derivative of the function is zero. It would be like trying to solve the equation x^2=3 by saying the derivative of x^2 must be zero. ie. 2x=0 so x=0. It makes no sense. PairTheBoard |
#6
|
|||
|
|||
Re: Does anyone have the math skills to isolate the variable, a.
Doesn't the logarithmic base here need to be specified?
|
#7
|
|||
|
|||
Re: Does anyone have the math skills to isolate the variable, a.
ln, is the natural log (base 2)
|
#8
|
|||
|
|||
Re: Does anyone have the math skills to isolate the variable, a.
[ QUOTE ]
ln, is the natural log (base 2) [/ QUOTE ] I assume you mean base e (2.71828.....) |
#9
|
|||
|
|||
Re: Does anyone have the math skills to isolate the variable, a.
yep, sorry about that.
|
#10
|
|||
|
|||
Re: Does anyone have the math skills to isolate the variable, a.
Can't anyone offer any suggestions on how variable, a, may be isolated.
|
|
|