Question

solve the equation for x. give an exact solution and also an approximate solution to four decimal places. 2^{2x}=12.7 a. the exact solution is x =

Explanation:

Step1: Take the natural - log of both sides

Take the natural logarithm of both sides of the equation $2^{2x}=12.7$. Using the property $\ln(a^b)=b\ln(a)$, we get $2x\ln(2)=\ln(12.7)$.

Step2: Solve for x

Divide both sides of the equation $2x\ln(2)=\ln(12.7)$ by $2\ln(2)$ to isolate $x$. So, $x = \frac{\ln(12.7)}{2\ln(2)}$.

Step3: Calculate the approximate value

Using a calculator, $\ln(12.7)\approx2.5427$ and $\ln(2)\approx0.6931$. Then $x=\frac{2.5427}{2\times0.6931}=\frac{2.5427}{1.3862}\approx1.8343$.

Answer:

The exact solution is $x=\frac{\ln(12.7)}{2\ln(2)}$, and the approximate solution to four - decimal places is $x\approx1.8343$.