Question

solve. round your answer to the nearest thousandth.\\(5^{4x - 3} = 8\\)\\(x = \square\\)

Explanation:

Step1: Take log of both sides

Take the natural logarithm (ln) of both sides of the equation \(5^{4x - 3}=8\). Using the property of logarithms \(\ln(a^b)=b\ln(a)\), we get:
\(\ln(5^{4x - 3})=\ln(8)\)
\((4x - 3)\ln(5)=\ln(8)\)

Step2: Solve for \(4x - 3\)

Divide both sides by \(\ln(5)\) to isolate \(4x - 3\):
\(4x - 3=\frac{\ln(8)}{\ln(5)}\)

Step3: Solve for \(4x\)

Add 3 to both sides of the equation:
\(4x=\frac{\ln(8)}{\ln(5)} + 3\)

Step4: Solve for \(x\)

Divide both sides by 4 to solve for \(x\):
\(x=\frac{\frac{\ln(8)}{\ln(5)} + 3}{4}\)

Now, calculate the numerical value. We know that \(\ln(8)\approx2.0794\) and \(\ln(5)\approx1.6094\).

First, calculate \(\frac{\ln(8)}{\ln(5)}\approx\frac{2.0794}{1.6094}\approx1.292\).

Then, \(\frac{\ln(8)}{\ln(5)} + 3\approx1.292 + 3 = 4.292\).

Finally, \(x=\frac{4.292}{4}\approx1.073\).

Answer:

\(1.073\)