Evaluate: log_{sqrt(10)}(x)^2+3log_{10}(x)=2-log_{0.1}(x)

Expression: ${\log_{ \sqrt{ 10 } }({ x })}^{2}+3\log_{ 10 }({ x })=2-\log_{ 0.1 }({ x })$

Determine the defined range

$\begin{array} { l }{\log_{ \sqrt{ 10 } }({ x })}^{2}+3\log_{ 10 }({ x })=2-\log_{ 0.1 }({ x }),& x > 0\end{array}$

Write the expression in exponential form with the base of $10$

${\log_{ {10}^{\frac{ 1 }{ 2 }} }({ x })}^{2}+3\log_{ 10 }({ x })=2-\log_{ 0.1 }({ x })$

Write the number in exponential form with the base of $10$

${\log_{ {10}^{\frac{ 1 }{ 2 }} }({ x })}^{2}+3\log_{ 10 }({ x })=2-\log_{ {10}^{-1} }({ x })$

Use $\log_{ {a}^{y} }({ b })=\frac{ 1 }{ y } \times \log_{ a }({ b })$ to transform the expression

${\left( 2\log_{ 10 }({ x }) \right)}^{2}+3\log_{ 10 }({ x })=2-\log_{ {10}^{-1} }({ x })$

Use $\log_{ {a}^{y} }({ b })=\frac{ 1 }{ y } \times \log_{ a }({ b })$ to transform the expression

${\left( 2\log_{ 10 }({ x }) \right)}^{2}+3\log_{ 10 }({ x })=2-\left( -\log_{ 10 }({ x }) \right)$

To raise a product to a power, raise each factor to that power

$4{\log_{ 10 }({ x })}^{2}+3\log_{ 10 }({ x })=2-\left( -\log_{ 10 }({ x }) \right)$

When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses

$4{\log_{ 10 }({ x })}^{2}+3\log_{ 10 }({ x })=2+\log_{ 10 }({ x })$

Move the expression to the left-hand side and change its sign

$4{\log_{ 10 }({ x })}^{2}+3\log_{ 10 }({ x })-\log_{ 10 }({ x })=2$

Collect like terms

$4{\log_{ 10 }({ x })}^{2}+2\log_{ 10 }({ x })=2$

To get an equation that is easier to solve, substitute $t$ for $\log_{ 10 }({ x })$

$4{t}^{2}+2t=2$

Solve the equation for $t$

$\begin{array} { l }t=-1,\\t=\frac{ 1 }{ 2 }\end{array}$

Substitute back $t=\log_{ 10 }({ x })$

$\begin{array} { l }\log_{ 10 }({ x })=-1,\\\log_{ 10 }({ x })=\frac{ 1 }{ 2 }\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=\frac{ 1 }{ 10 },\\\log_{ 10 }({ x })=\frac{ 1 }{ 2 }\end{array}$

Solve the equation for $x$

$\begin{array} { l }\begin{array} { l }x=\frac{ 1 }{ 10 },\\x=\sqrt{ 10 }\end{array},& x > 0\end{array}$

Check if the solution is in the defined range

$\begin{array} { l }x=\frac{ 1 }{ 10 },\\x=\sqrt{ 10 }\end{array}$

The equation has $2$ solutions

$\begin{align*}&\begin{array} { l }x_1=\frac{ 1 }{ 10 },& x_2=\sqrt{ 10 }\end{array} \\&\begin{array} { l }x_1=0.1,& x_2\approx3.16228\end{array}\end{align*}$