i have

6.644 = -ln[ln(T/T-1)]

and i need to find a value of t

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- Jan 12th 2007, 10:38 AMquestionanyone help???
i have

6.644 = -ln[ln(T/T-1)]

and i need to find a value of t - Jan 12th 2007, 10:58 AMearboth
- Jan 12th 2007, 11:09 AMtopsquark
You're looking for t or T? (This isn't really as stupid a question as it sounds. Technically they are different variables, though here the meaning is clear enough.)

$\displaystyle 6.644 = -ln \left [ ln \left ( \frac{T}{T-1} \right ) \right ] $

$\displaystyle -6.644 = ln \left [ ln \left ( \frac{T}{T-1} \right ) \right ] $

$\displaystyle e^{-6.644} = ln \left ( \frac{T}{T-1} \right ) $

Ordinarily I would leave the LHS alone, but as the exponent is in decimal form and we are going to have a complicated form in the end, it is just easier to get a decimal expression. Note: I am going to keep a ridiculous number of digits in the expression until the final answer to make rounding errors as small as reasonable.

$\displaystyle 0.001301809877 = ln \left ( \frac{T}{T-1} \right ) $

$\displaystyle e^{0.001301809877} = \frac{T}{T-1}$

Again going to the decimal expression:

$\displaystyle 1.0013026573 = \frac{T}{T-1}$

$\displaystyle 1.0013026573 (T - 1) = T$

$\displaystyle 1.0013026573T - 1.0013026573 = T$

$\displaystyle 0.0013026573T = 1.0013026573$

$\displaystyle T = \frac{1.0013026573}{0.0013026573} \approx 768.662$

If you prefer the exact expression is got the same way and is:

$\displaystyle t = \frac{e^{e^{-6.644}}}{e^{e^{-6.644}} - 1}$

-Dan