Solving $(1.44)^t=t^{1.44}$ I've been trying to solve the equation  $(1.44)^t=t^{1.44}$, but other than the obvious solution ($t=1.44$) I haven't had much luck manipulating this into something useful. By taking the log of both sides I'm able to get $\dfrac{\ln t}{t}=\dfrac{\ln 1.44}{1.44}$, but then I'm left with essentially the same problem--my variables are in two different "places" and I can't figure out how to combine them. I can also use exponents to rewrite this as $t^{\frac{1}{t}}=e^{\frac{\ln 1.44}{1.44}}$ or $t^{t^{-1}}=e^{\frac{\ln 1.44}{1.44}}$.

I also tried rewriting the equation as $(1.44)^t-t^{1.44}=0$ and factoring out $1.44-t$, but it quickly turned into a complete mess. Any thoughts on how to approach this? I know two other solutions exist by looking at the graphs of the two functions.
One 'obvious' solution is of course $t=1.44$ for any other solutions you need to use a numerical method, because as you realized too, you cannot solve the equation for $t$ analitically.
Here is a plot. Showing the two curves (there is no real value for $t^1.44$ when $t&lt;0$, so that curve is only plotted for positive $t$.



You can also take logs. Here is the plot for that. 

Unfortunately, I cannot think of any way of getting the exact value of the larger solution in terms of elementary functions (or even, at the moment, "special" functions).

[Added later] Just to be clear, Mike's solution (at 0839Z 12 Sep 14) is not an analytic solution, although there may be one if someone could just think of it.