Question

solve. round your answer to the nearest thousandth. 38 = 3^x x =

Explanation:

Step1: Take logarithm on both sides

To solve for \( x \) in the equation \( 38 = 3^{x} \), we take the natural logarithm (ln) of both sides. This gives us \( \ln(38)=\ln(3^{x}) \).

Step2: Use logarithm power rule

Using the power rule of logarithms, \( \ln(a^{b}) = b\ln(a) \), so the right - hand side becomes \( x\ln(3) \). Now our equation is \( \ln(38)=x\ln(3) \).

Step3: Solve for x

To isolate \( x \), we divide both sides of the equation by \( \ln(3) \). So \( x = \frac{\ln(38)}{\ln(3)} \).

Step4: Calculate the value

We know that \( \ln(38)\approx3.637586 \) and \( \ln(3)\approx1.098612 \). Then \( x=\frac{3.637586}{1.098612}\approx3.311 \) (rounded to the nearest thousandth).

Answer:

\( x\approx3.311 \)