Question

solve the equation. give an exact solution, and also an approximate solution to four

$e^{9x}=6$

a. the exact solution is $x = \frac{e^{6}}{9}$

Explanation:

Step1: Apply natural - logarithm to both sides

Take the natural logarithm of both sides of the equation $e^{9x}=6$. We know that $\ln(e^{a}) = a$, so $\ln(e^{9x})=\ln(6)$. Then $9x=\ln(6)$.

Step2: Solve for x

Divide both sides of the equation $9x = \ln(6)$ by 9. We get $x=\frac{\ln(6)}{9}$.

Step3: Calculate the approximate value

Using a calculator, $\ln(6)\approx1.7918$ and $\frac{\ln(6)}{9}\approx\frac{1.7918}{9}\approx0.1991$.

Answer:

The exact solution is $x = \frac{\ln(6)}{9}$, and the approximate solution to four - decimal places is $x\approx0.1991$