$${\mathtt{1\,000}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{4}}} = {\frac{{\mathtt{1}}}{{{\mathtt{x}}}^{{log}_{10}\left({\mathtt{x}}\right)}}}$$
+33614 Write this as
$$\\
1000\times x^4=x^{-\log_{10}x}\\\\
\text{Take logs of both sides and use properties of logs to get}\\\\
\log_{10}(1000)+\log_{10}(x^4)=\log_{10}(x^{-\log_{10}(x)})\\\\
3+4\log_{10}(x)=-(\log_{10}(x))^2\\\\
\text{Rearrange}\\\\
\log_{10}(x))^2+4\log_{10}(x)+3=0\\\\
\text{This is a quadratic in log(x) and factors nicely}\\\\
(log_{10}(x)+1)(log_{10}(x)+3)=0\\\\
\text{so the two solutions for x are}\\\\
log_{10}(x)=-1\quad x=10^{-1}=0.1\\
log_{10}(x)=-3\quad x = 10^{-3}=0.001$$
.
+33614 Write this as
$$\\
1000\times x^4=x^{-\log_{10}x}\\\\
\text{Take logs of both sides and use properties of logs to get}\\\\
\log_{10}(1000)+\log_{10}(x^4)=\log_{10}(x^{-\log_{10}(x)})\\\\
3+4\log_{10}(x)=-(\log_{10}(x))^2\\\\
\text{Rearrange}\\\\
\log_{10}(x))^2+4\log_{10}(x)+3=0\\\\
\text{This is a quadratic in log(x) and factors nicely}\\\\
(log_{10}(x)+1)(log_{10}(x)+3)=0\\\\
\text{so the two solutions for x are}\\\\
log_{10}(x)=-1\quad x=10^{-1}=0.1\\
log_{10}(x)=-3\quad x = 10^{-3}=0.001$$
.
