Question

solve. round your answer to the nearest thousandth.\\(3^x = 31\\)\\(x = \square\\)

Explanation:

Step1: Take log of both sides

To solve for \( x \) in the equation \( 3^x = 31 \), we take the logarithm of both sides. Using the natural logarithm (ln) for this, we get:
\( \ln(3^x) = \ln(31) \)

Step2: Apply logarithm power rule

By the power rule of logarithms, \( \ln(a^b)=b\ln(a) \), so the left side becomes:
\( x\ln(3) = \ln(31) \)

Step3: Solve for x

Divide both sides by \( \ln(3) \) to isolate \( x \):
\( x = \frac{\ln(31)}{\ln(3)} \)

Step4: Calculate the value

Using a calculator, we find \( \ln(31) \approx 3.433987 \) and \( \ln(3) \approx 1.098612 \). Then:
\( x \approx \frac{3.433987}{1.098612} \approx 3.125 \) (rounded to the nearest thousandth)

Answer:

\( 3.125 \)