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?
$\endgroup$7 Answers
$\begingroup$Hint: Take the log of both sides. You will get a quadratic equation in $\log x$. The equation is even "nice."
$\endgroup$ $\begingroup$Hint: Apply log on both side and try to solve
$\endgroup$ $\begingroup$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.
$\endgroup$ 3 $\begingroup$$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$
$\endgroup$ $\begingroup$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 $x=10$
$\endgroup$ $\begingroup$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.
$\endgroup$ 7 $\begingroup$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.
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