91Ó°ÊÓ

First, rewrite the equation as: $$ \begin{aligned} (\log (3 x)) \log x &= 4 \end{aligned} $$ Now, exponentiate both sides with base 10 to rewrite the equation in terms of exponentials: $$ \begin{aligned} 10^{(\log (3 x)) \log x}& = 10^4 \end{aligned} $$

Use the properties of exponents to simplify the equation further. An important property to remember is \(a^{bc}=(a^b)^c\). So applying this, we get: $$ \begin{aligned} (10^{\log (3 x)})^{\log x}& = 10^4 \end{aligned} $$ Since \(10^{\log y} = y\), the equation becomes: $$ \begin{aligned} (3x)^{\log x}=10^4 \end{aligned} $$ It is still not easy to isolate x in this form, so let's take another approach and apply \(\log (y^x)=x \cdot \log y\) on both sides. After taking logarithm with base 10 on both sides, we get: $$ \begin{aligned} \log (3x^{\log x}) &= \log(10^4) \end{aligned} $$ Now, apply the mentioned logarithm property: $$ \begin{aligned} \log x(\log (3x)) &= 4 \end{aligned} $$ The left side of the equation is the same as our original equation.

Looking back at our original equation \((\log (3 x)) \log x=4\) and this new equation \(\log x(\log (3x)) = 4\), it is clear that we haven't made much progress with transforming the equation into a simpler form. However, we can proceed with trial and error to find potential solutions. If \(x = 10\): $$ \begin{aligned} (\log (3 \times 10)) \log 10 = (\log 30) (1) = \log 10^4 = 4 \end{aligned} $$ This matches our equation. Since logarithmic functions are continuous and strictly increasing, a single match means there are no other solutions. Hence, the only value for x that satisfies the given equation is: $$ x = 10 $$