Answer:
x = 10 or x = 100
Step-by-step explanation:
The domain of the equation: x > 0.
We have:
[tex]2+(\log x)^2=3\log x[/tex]
Substitute [tex]\log x=t[/tex]:
[tex]2+t^2=3t[/tex] subtract 3t from both sides
[tex]t^2-3t+2=0[/tex]
[tex]t^2-2t-t+2=0[/tex]
[tex]t(t-2)-1(t-2)=0[/tex]
[tex](t-2)(t-1)=0\iff t-2=0\ \vee\ t-1=0[/tex]
[tex]t-2=0[/tex] add 2 to both sides
[tex]t=2[/tex]
[tex]t-1=0[/tex] add 1 to both sides
[tex]t=1[/tex]
We return to substitution:
[tex]t=2\to\log x=2[/tex]
and
[tex]t=1\to\log x=1[/tex]
Use
[tex]\log_ab=c\iff a^c=b[/tex]
[tex]\log a=\log_{10}a[/tex]
[tex]\log_aa=1[/tex]
[tex]\log x=2\iff x=10^2\to x=100\\\\\log x=1\to x=10[/tex]
