Answer
The solution set is $\left \{\pm \sqrt {\frac{\ln (45)}{\ln(3)}}\approx \pm 1.86 \right \}$.
Work Step by Step
The given equation is
$3^{x^{2}}=45$
Take the natural logarithm on both sides.
$\ln (3^{x^{2}})=\ln (45)$
Use the power rule.
$x^2 \ln (3)=\ln (45)$
Divide both sides by $\ln(3)$.
$\frac{x^2 \ln (3)}{\ln(3)}=\frac{\ln (45)}{\ln(3)}$
$x^2=\frac{\ln (45)}{\ln(3)}$
Take the square root on both sides.
$\sqrt {x^2}=\pm \sqrt {\frac{\ln (45)}{\ln(3)}}$
$x=\pm \sqrt {\frac{\ln (45)}{\ln(3)}}$
$x=\pm 1.86$
