Question

solve. round your answer to the nearest thousandth.\\(3^{4x + 3} = 7\\)\\(x = \square\\)submit

Explanation:

Step1: Take logarithm on both sides

To solve the equation \(3^{4x + 3}=7\), we can take the natural logarithm (ln) on both sides. According to the logarithm property \(\ln(a^b)=b\ln(a)\), we have:
\(\ln(3^{4x + 3})=\ln(7)\)
\((4x + 3)\ln(3)=\ln(7)\)

Step2: Solve for \(x\)

First, divide both sides of the equation \((4x + 3)\ln(3)=\ln(7)\) by \(\ln(3)\):
\(4x+3 = \frac{\ln(7)}{\ln(3)}\)
Then, subtract 3 from both sides:
\(4x=\frac{\ln(7)}{\ln(3)}-3\)
Finally, divide both sides by 4:
\(x=\frac{\frac{\ln(7)}{\ln(3)}-3}{4}\)

We know that \(\ln(7)\approx1.9459\), \(\ln(3)\approx1.0986\), so \(\frac{\ln(7)}{\ln(3)}\approx\frac{1.9459}{1.0986}\approx1.7712\)
Then \(\frac{\ln(7)}{\ln(3)}-3\approx1.7712 - 3=- 1.2288\)
And \(x=\frac{-1.2288}{4}\approx - 0.307\)

Answer:

\(x\approx - 0.307\)