What is the sum of the real solutions to x^(logx) = (x^3)/100 ?

Question

anonymous · Answer

first take the log of both sides to get the variable out of the exponent.

$$log(x^{log(x)})=log(\frac{x^3}{100})$$
then use the properties of logs so put everything on the ground floor.

$$log(x)log(x)=log(x^3)-log(100)=3log(x)-2$$

hope it is clear than log(100)=2 since 10^2=100

$$(log(x))^2=3log(x)-2$$
now you have a quadratic equation (in log(x) so set = 0 and solve
$$(log(x))^2-3log(x)+2=0$$
this one factors.
$$(log(x)-2)(log(x)-1)=0$$

two solutions are
$$log(x)=2$$
and
$$log(x)=1$$
now find x
$$x=10^1$$
$$x=10^2=100$$
total is 110

anonymous · Answer

Thanks, I remember learning this in the beginning of the year now!