Solving $x^{\log(x)}=\frac{x^3}{100}$

Question

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?

Answer 1

Hint: Apply log on both side and try to solve

Answer 2

Hint: Take the log of both sides. You will get a quadratic equation in $\log x$. The equation is even "nice."

Answer 3

There are no solutions in the real numbers.

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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.

Answer 4

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.

Answer 5

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 $x=10$

Answer 6

$x^{\log(x)}=\frac{x^3}{100}$

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$

Answer 7

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.