Equation involving logarithm, solvable without calculator? I'd like to know if I can solve the following equation without calculator:
$(0.4)^t=5t$
I don't think it's possible, cause I always get stuck on formulas of the form $e^t=t$ or $t=\ln t$
I've also put the equation into wolframalpha, which was of no use to me unfortunately.
I'm not interested in the answer containing a W-function.
Just want to know whether I can find the real solution or not!
Thanks! 
 A: No, it is not possible to solve in terms of elementary functions.  That is why W|A returns an answer using the W function.
When you find the formula boils down to $t=\ln t$, you've come across a form of Lambert's Transcendental Equation: 
A: You will need a numerical method or some not-so-obvious special function to do this.
A: There is a solution $t$, since $(0.4)^t\gt 5t$ at $t=0$, and $(0.4)^t\lt 5t$ at $t=1$. It is easy to see that there cannot be a solution outside the interval $(0,1)$. 
There is no rational solution. For suppose that $t=\frac{p}{q}$ is a solution, where $p$ and $q$ are relatively prime integers, neither equal to $0$. Since $0\lt t\lt 1$, we have $p\lt q$. Then from
$$\left(\frac{2}{5}\right)^{p/q}=5\frac{p}{q}$$
we obtain 
$$2^pq^q=5^{p+q}p^q.$$
Thus $p$ is a power of $2$. This is impossible, since then $2^p \lt p^q$. 
By using the Gelfond-Schneider theorem, we can now prove that $t$ cannot be an irrational algebraic number. For if $t$ is irrational algebraic, then $(0.4)^t$ is transcendental, but $5t$ is not.
So $t$ is transcendental. We have not ruled out the possibility that $t$ is a simple combination of "simple" transcendentals, such as $\log 2$, $\log 3$, \sin 1$, and so on.  
