Question

solve the equation for x. give an exact solution and also an approximate solution to four decimal places. 5^{2x}=26.8
a. the exact solution is x=

Explanation:

Step1: Take natural logarithm of both sides

Take the natural - logarithm of $5^{2x}=26.8$. Using the property $\ln(a^b)=b\ln(a)$, we get $2x\ln(5)=\ln(26.8)$.

Step2: Solve for x

Divide both sides of the equation $2x\ln(5)=\ln(26.8)$ by $2\ln(5)$. So, $x = \frac{\ln(26.8)}{2\ln(5)}$.

Step3: Calculate the approximate value

Using a calculator, $\ln(26.8)\approx3.2909$ and $\ln(5)\approx1.6094$. Then $x=\frac{3.2909}{2\times1.6094}=\frac{3.2909}{3.2188}\approx1.0224$.

Answer:

The exact solution is $x = \frac{\ln(26.8)}{2\ln(5)}$, and the approximate solution to four decimal places is $x\approx1.0224$.