Solve the following equation:
$x^{\log x}=1000 x^{2}$

Hint: In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since $1000=10 \times 10 \times 10=10^{3}$, the "logarithm base 10 " of 1000 is 3 , or $\log _{10}(1000)=3$. The logarithm of x to base b is denoted as $\log _{b}(x)$, or without parentheses, $\log _{b} x$, or even without the explicit base, $\log x$, when no confusion is possible, or when the base does not matter such as in $\log _{10}(1000)=3$.

Complete step-by-step answer:
The general case is when you raise a number b to the power of y to get x:
$b^{y}=x$
The number b is referred to as the base of this expression. The base is the number that is raised to a particular power-in the above example, the base of the expression $2^{3}=8$ is 2. It is easy to make the base the subject of the expression: all you have to do is take the y-th root of both sides. This gives:
$b=\sqrt[y]{x}$
It is less easy to make the subject of the expression. Logarithms allow us to do this:
$y=\log _{b} x$
This expression means that y is equal to the power that you would raise b to, to get x. This operation undoes exponentiation because the logarithm of $x$ tells you the exponent that the base has been raised to.
$x^{\log x}=1000 x^{2}$
$\Rightarrow \log \mathrm{x}(\log \mathrm{x})=\log (1000)+\log \mathrm{x}^{2}$
$\Rightarrow(\log \mathrm{x})^{2}=3+2 \log \mathrm{x}$
$\Rightarrow(\log \mathrm{x})^{2}-2 \log \mathrm{x}-3=0$
$\Rightarrow(\log \mathrm{x})^{2}-3 \log \mathrm{x}+\log \mathrm{x}-3=0$
$\Rightarrow(\log \mathrm{x}+1)(\log \mathrm{x}-3)=0$
$\log _{\mathrm{X}}=-1$ or 3
$\mathrm{x}=\dfrac{1}{10}$ or 1000.

Note: The logarithm base 10 (that is $b=10$ ) is called the common logarithm and is commonly used in science and engineering. The natural logarithm has the number e (that is $b \approx 2.718$ ) as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative. The binary logarithm uses base 2 (that is $b=2$ ) and is commonly used in computer science. Logarithms are examples of concave functions.