Questions: Use logarithms to solve (9 e^2 x=5). If there is more than one solution, enter them as a comma separated list. If there is no solution, enter "NONE". (x=)

Solution Steps

To solve the equation \(9 e^{2x} = 5\) using logarithms, follow these steps:

1. Isolate the exponential term by dividing both sides by 9.
2. Take the natural logarithm (ln) of both sides to remove the exponential.
3. Solve for \(x\) by isolating it on one side of the equation.

Starting with the equation: \[ 9 e^{2x} = 5 \] we divide both sides by 9 to isolate the exponential term: \[ e^{2x} = \frac{5}{9} \]

Next, we take the natural logarithm of both sides: \[ \ln(e^{2x}) = \ln\left(\frac{5}{9}\right) \]

Using the property \(\ln(e^y) = y\), we simplify the left side: \[ 2x = \ln\left(\frac{5}{9}\right) \]

Now, we solve for \(x\) by dividing both sides by 2: \[ x = \frac{1}{2} \ln\left(\frac{5}{9}\right) \] Calculating the value gives: \[ x \approx -0.2939 \]

Final Answer