Question

solve the following equation.
$3^{x^2} = 47$
the solution set is
(type an exact answer, using radicals as needed.)

Explanation:

Step1: Take log of both sides

Take the natural logarithm (ln) of both sides of the equation \(3^{x^2}=47\). We get \(\ln(3^{x^2})=\ln(47)\).

Step2: Use logarithm power rule

Using the power rule of logarithms \(\ln(a^b)=b\ln(a)\), the left - hand side becomes \(x^2\ln(3)=\ln(47)\).

Step3: Solve for \(x^2\)

Divide both sides of the equation \(x^2\ln(3)=\ln(47)\) by \(\ln(3)\) to isolate \(x^2\). So, \(x^2 = \frac{\ln(47)}{\ln(3)}\).

Step4: Solve for \(x\)

Take the square root of both sides. Since \(x^2=\frac{\ln(47)}{\ln(3)}\), then \(x=\pm\sqrt{\frac{\ln(47)}{\ln(3)}}\) (or we can also write it as \(x = \pm\frac{\sqrt{\ln(47)}}{\sqrt{\ln(3)}}\) by rationalizing the square root in the fraction, but the first form is also correct).

Answer:

\(\pm\sqrt{\frac{\ln(47)}{\ln(3)}}\) (or equivalent forms like \(\pm\frac{\sqrt{\ln(47)}}{\sqrt{\ln(3)}}\))