The discussion revolves around manipulating an equation involving a logarithmic expression to achieve a desired form. The original equation includes a natural logarithm and is related to a physical context, likely involving fluid dynamics or thermodynamics.
Participants are exploring different mathematical approaches to manipulate the equation. Some guidance has been offered regarding the use of approximations and properties of logarithms, but there is no explicit consensus on the method to be used.
There is a mention of the approximation ##\ln(1+a) \approx a## being valid for small values of ##a##, which is a key point under discussion. The conversation also touches on the Taylor series expansion of the logarithm, indicating a focus on mathematical foundations.
shreddinglicks said:Homework Statement
I want to manipulate an equation to suit a desired form.
Homework Equations
##(-h^2/2uc)*(dp/dx)*ln((1+c*(x/h)^2)/(1+c))##
becomes
##-(h^2/2u)*(dp/dx)*(1-(x/h)^2)##
The Attempt at a Solution
I have no idea, I'm not even sure how the natural log disappears. [/B]
Dick said:They are using the approximation ##\ln(1+a) \approx a## valid when ##a## is small. Try that along with other rules about logs.
shreddinglicks said:Is there a link to somewhere online showing that approximation you gave? Just curious.
Dick said:I don't know any good links. But approximations like this generally come from taking the first terms of the Taylor series. ##\ln(1+x)=x-
\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots## Keeping just the first term gives the approximation. Similarly, ##\sin(x) \approx x## etc.