Solving for a Variable as an Exponent

samhus

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Oct 23, 2014
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Hi!

I'm trying to understand how an equation was solved but am having some trouble.

The equation is .9 = e ^ (-.0010362 * T)

The formula is transposed to T = -1,000 ln (.9) which results in the answer of T = 109.2

I'm just trying to understand the steps required in transposing the original equation to the new one.

Any help would be greatly appreciated!
 
First, you should understand that "1000 ln(.9)" is approximately equal to T but that is NOT the exact answer.

To solve an equation for a specific variable, "undo" whatever is done to that variable. That means applying the "inverse" function to each function in turn.

Here, in \(\displaystyle e ^{-.0010362 T}= .9\), two things have been done to T. First, it has been multiplied by -.0010362, second, the exponential function, \(\displaystyle exp(x)= e^x\) has been applied. We apply the inverse of both of those in the reverse order and, of course, whatever we do to one side of the equation we must do to the other.

The inverse of the exponential is the natural logarithm:
\(\displaystyle ln(e^{-.0010362T}= -.0010362T= ln(.9)\).

The inverse of "multiply by -.0010362" is "divide by -0.010362":
\(\displaystyle \frac{-.0010362T}{-.0010362}= T= \frac{ln(.9)}{-.0010362}\)

\(\displaystyle \frac{1}{-.0010362}= 965.06465933217525574213472302644,\) not 1000.

If the original problem were \(\displaystyle e^{-.001 T}= .9\), THEN the solution would be T= 1000 ln(.9).
 
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