Simple Algebra Problem

**sentimentGX4** · September 16th 2012, 12:56 PM

I need to solve this algebra problem for a part of a question in my Algorithms class but I have no idea how. I have access to a TI-84.

10000*x(ln(x)/ln(2))= 10¹²

How could I go about solving this? What identities can I use to simplify this? Help?

**skeeter** · September 16th 2012, 01:05 PM

you'll need to use the calculator, can't be done by hand using elementary algebraic methods.

note that this equation can be simplified first ...

$10^5 \cdot \frac{x\ln{x}}{\ln{2}} = 10^{12}$

$\frac{x\ln{x}}{\ln{2}} = 10^7$

now type the expression

$\frac{x\ln{x}}{\ln{2}} - 10^7$

into Y1 and use the "solver" feature to find where the equation = 0

**HallsofIvy** · September 16th 2012, 01:32 PM

This isn't an "elementary algebraic method" but from $\frac{x ln(x)}{ln(2)}= 10^7$ we easily get $xln(x)= 10^7ln(2)= ln(2^{10^7})$ and, taking the exponential of both sides $e^xe^{ln(x)}= xe^x= e^{ln(2^{10^7})}= 2^{10^7}$ . Now, we apply the "Lambert W function" (which is defined as the inverse function to $f(x)= xe^x$ ) we have $x= W(2^{10^7})$ .

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