Question

solve. round your answer to the nearest thousandth.\\(4^{2x} = 7\\)\\(x = \square\\)

Explanation:

Step1: Take log of both sides

Take the natural logarithm (ln) of both sides of the equation \(4^{2x} = 7\). We get \(\ln(4^{2x})=\ln(7)\).

Step2: Use logarithm power rule

Using the power rule of logarithms \(\ln(a^b)=b\ln(a)\), the left - hand side becomes \(2x\ln(4)=\ln(7)\).

Step3: Solve for x

First, we can rewrite \(\ln(4)\) as \(2\ln(2)\), so the equation is \(2x\times2\ln(2)=\ln(7)\), or \(4x\ln(2)=\ln(7)\). Then, solve for \(x\): \(x = \frac{\ln(7)}{4\ln(2)}\). Alternatively, from \(2x\ln(4)=\ln(7)\), we can also solve for \(x\) as \(x=\frac{\ln(7)}{2\ln(4)}\).
We know that \(\ln(7)\approx1.9459\) and \(\ln(4) = 2\ln(2)\approx1.3863\).
Substitute these values into the formula \(x=\frac{\ln(7)}{2\ln(4)}\):
\(x=\frac{1.9459}{2\times1.3863}=\frac{1.9459}{2.7726}\approx0.702\)

Answer:

\(0.702\)