Solving $x^{\log(x)}=\frac{x^3}{100}$ How do I find the solution to:
$$x^{\log(x)}=\frac{x^3}{100}$$
So I multiplied 100 both sides getting:
$$100x^{\log(x)}=x^3$$
Now what should I do?
 A: Hint: Take the log of both sides. You will get a quadratic equation in $\log x$.
 The equation is even "nice."
A: Hint: Apply log on both side and try to solve
A: I suppose $\log$ means $\log_{10}$? I'm not familiar with this sort of notation.
Take logarithm on both sides, and you will get $2+\log^2x=3\log x$. Substitute $\log x$ with t. And you get $t^2-3t+2=0$, therefore $(t-1)(t-2)=0$. That should do it.
A: $x^{\log(x)}=\frac{x^3}{100}$ 
Taking log on both sides you get : 
$log(x) log(x) = log(\frac{x^3}{100})$ = log(x) log(x) = 3logx - 2log10 = 3logx -2 
$\Rightarrow (log(x))^2  = 3logx -2 $ 
Now putting log(x) = t 
$\Rightarrow t^2=3t-2$
Now you can solve for t as this is a quadratic in t. you get (t-2)(t-1) $\Rightarrow t = 2 ; t = 1$ 
$\Rightarrow logx = 2  \Rightarrow x = 100 $ ; and $ logx = 1  \Rightarrow x = 10$
A: There are no solutions in the real numbers.

Edit: If, as RecklessReckoner suggests, the question was misstated and the intent was to use $\log_{10},$ then the solution can be found easily by taking logarithms.
A: Since $(\log(x))^2=\log (x^{\log x})=\log (x^3/100)=3\log(x)-2$, we have $(\log(x))^2-3\log(x)+2=0$. Hence, $\log(x)=2$ and $\log(x)=1$. Therefore, $x=100$ atau $x=10$
A: I am entering an answer just to point out a peculiarity of this equation, and the importance of interpreting the exponent "correctly". (This should also clarify  Charles' answer.)  I have graphed the quadratic functions $ \ (\ln x)^2 - (3 \ln x) + (\ln 100) \ $ in blue and $ \ (\log x)^2 - (3 \log x) + 2 \ $ in red.  Notice that the blue curve has no x-intercepts; the quadratic equation for natural logarithms yields a negative discriminant since $ \ (-3)^2 < 4 \ln 100 \ $.  So common logarithms are intended in this problem.


I had to use two graphs because the graph of the common logarithmic function is very shallow, but it does cross the x-axis at 100.
