Question

Solve the value of x; 2+(㏒x)^2=3 ㏒x

Answer 1

Answer:

x = 10 or x = 100

Step-by-step explanation:

The domain of the equation: x > 0.

We have:

$2+(\log x)^2=3\log x$

Substitute $\log x=t$:

$2+t^2=3t$         subtract 3t from both sides

$t^2-3t+2=0$

$t^2-2t-t+2=0$

$t(t-2)-1(t-2)=0$

$(t-2)(t-1)=0\iff t-2=0\ \vee\ t-1=0$

$t-2=0$          add 2 to both sides

$t=2$

$t-1=0$      add 1 to both sides

$t=1$

We return to substitution:

$t=2\to\log x=2$

and

$t=1\to\log x=1$

Use

$\log_ab=c\iff a^c=b$

$\log a=\log_{10}a$

$\log_aa=1$

$\log x=2\iff x=10^2\to x=100\\\\\log x=1\to x=10$