Question

solve for x.
\\(9^{8x} = 4^{-x - 3}\\)
round your answer to the nearest thousandth.
do not round any intermediate computations.
\\(x = \square\\)

Explanation:

Step1: Take natural log of both sides

Take the natural logarithm of both sides of the equation \(9^{8x} = 4^{-x - 3}\). Using the logarithm power rule \(\ln(a^b)=b\ln(a)\), we get:
\(8x\ln(9)=(-x - 3)\ln(4)\)

Step2: Expand the right - hand side

Expand the right - hand side of the equation:
\(8x\ln(9)=-x\ln(4)-3\ln(4)\)

Step3: Move all terms with x to the left - hand side

Add \(x\ln(4)\) to both sides of the equation to get all terms with \(x\) on the left - hand side:
\(8x\ln(9)+x\ln(4)=- 3\ln(4)\)

Step4: Factor out x

Factor out \(x\) from the left - hand side of the equation:
\(x(8\ln(9)+\ln(4))=-3\ln(4)\)

Step5: Solve for x

Now, solve for \(x\) by dividing both sides of the equation by \((8\ln(9)+\ln(4))\):
\(x=\frac{-3\ln(4)}{8\ln(9)+\ln(4)}\)

We know that \(\ln(9)=\ln(3^2) = 2\ln(3)\approx2\times1.0986 = 2.1972\), \(\ln(4)=\ln(2^2)=2\ln(2)\approx2\times0.6931 = 1.3862\)

Substitute these values into the formula for \(x\):

First, calculate the denominator: \(8\ln(9)+\ln(4)=8\times2.1972 + 1.3862=17.5776+1.3862 = 18.9638\)

Then, calculate the numerator: \(-3\ln(4)=-3\times1.3862=-4.1586\)

Now, \(x=\frac{-4.1586}{18.9638}\approx - 0.219\)

Answer:

\(x\approx - 0.219\)