Question

question
solve for ( x ), rounding to the nearest hundredth.
( 3^{3x} = 94 )

Explanation:

Step1: Take log of both sides

Take the natural logarithm (ln) of both sides of the equation \(3^{3x} = 94\). Using the logarithm power rule \(\ln(a^b)=b\ln(a)\), we get:
\(\ln(3^{3x})=\ln(94)\)
\(3x\ln(3)=\ln(94)\)

Step2: Solve for x

First, isolate \(x\) by dividing both sides by \(3\ln(3)\):
\(x=\frac{\ln(94)}{3\ln(3)}\)
Now, calculate the values of the logarithms. We know that \(\ln(94)\approx4.5433\) and \(\ln(3)\approx1.0986\). Substitute these values into the formula:
\(x=\frac{4.5433}{3\times1.0986}\)
\(x=\frac{4.5433}{3.2958}\)
\(x\approx1.38\) (rounded to the nearest hundredth)

Answer:

\(x\approx1.38\)