(1 point) Find all solutions \( x \) to each of the following equations. Separate multiple answers with commas. Use no solution or none if there are no solutions. \( 3 e^{2 x}=5 \) has solutions \( x=\square \) \( 10 e^{2 x}=7 e^{2 x}+5 \) has solutions \( x=\square \) \( 10 e^{2 x}=7 e^{2 x}+35 \) has solutions \( x=\square \) \( \frac{e^{2 x}}{e^{2 x}+5}=\frac{7}{10} \) has solutions \( x=\square \)

Let's solve each equation step by step. 1. **Equation:** \( 3 e^{2 x} = 5 \) To solve for \( x \), we first isolate \( e^{2x} \): \[ e^{2x} = \frac{5}{3} \] Now, take the natural logarithm of both sides: \[ 2x = \ln\left(\frac{5}{3}\right) \] Divide by 2: \[ x = \frac{1}{2} \ln\left(\frac{5}{3}\right) \] **Solution:** \( x = \frac{1}{2} \ln\left(\frac{5}{3}\right) \) 2. **Equation:** \( 10 e^{2 x} = 7 e^{2 x} + 5 \) Rearranging gives: \[ 10 e^{2x} - 7 e^{2x} = 5 \] Simplifying: \[ 3 e^{2x} = 5 \] This is the same as the first equation, so we already found the solution: \[ e^{2x} = \frac{5}{3} \] Thus: \[ x = \frac{1}{2} \ln\left(\frac{5}{3}\right) \] **Solution:** \( x = \frac{1}{2} \ln\left(\frac{5}{3}\right) \) 3. **Equation:** \( 10 e^{2 x} = 7 e^{2 x} + 35 \) Rearranging gives: \[ 10 e^{2x} - 7 e^{2x} = 35 \] Simplifying: \[ 3 e^{2x} = 35 \] Now, isolate \( e^{2x} \): \[ e^{2x} = \frac{35}{3} \] Taking the natural logarithm: \[ 2x = \ln\left(\frac{35}{3}\right) \] Divide by 2: \[ x = \frac{1}{2} \ln\left(\frac{35}{3}\right) \] **Solution:** \( x = \frac{1}{2} \ln\left(\frac{35}{3}\right) \) 4. **Equation:** \( \frac{e^{2 x}}{e^{2 x}+5} = \frac{7}{10} \) Cross-multiplying gives: \[ 10 e^{2x} = 7(e^{2x} + 5) \] Expanding the right side: \[ 10 e^{2x} = 7 e^{2x} + 35 \] Rearranging gives: \[ 10 e^{2x} - 7 e^{2x} = 35 \] Simplifying: \[ 3 e^{2x} = 35 \] Isolating \( e^{2x} \): \[ e^{2x} = \frac{35}{3} \] Taking the natural logarithm: \[ 2x = \ln\left(\frac{35}{3}\right) \] Divide by 2: \[ x = \frac{1}{2} \ln\left(\frac{35}{3}\right) \] **Solution:** \( x = \frac{1}{2} \ln\left(\frac{35}{3}\right) \) Now, summarizing all solutions: - \( 3 e^{2 x} = 5 \) has solutions \( x = \frac{1}{2} \ln\left(\frac{5}{3}\right) \) - \( 10 e^{2 x} = 7 e^{2 x} + 5 \) has solutions \( x = \frac{1}{2} \ln\left(\frac{5}{3}\right) \) - \( 10 e^{2 x} = 7 e^{2 x} + 35 \) has solutions \( x = \frac{1}{2} \ln\left(\frac{35}{3}\right) \) - \( \frac{e^{2 x}}{e^{2 x}+5} = \frac{7}{10} \) has solutions \( x = \frac{1}{2} \ln\left(\frac{35}{3}\right) \) Final answers: - \( 3 e^{2 x} = 5 \) has solutions \( x = \frac{1}{2} \ln\left(\frac{5}{3}\right) \) - \( 10 e^{2 x} = 7 e^{2 x} + 5 \) has solutions \( x = \frac{1}{2} \ln\left(\frac{5}{3}\right) \) - \( 10 e^{2 x} = 7 e^{2 x} + 35 \) has solutions \( x = \frac{1}{2} \ln\left(\frac{35}{3}\right) \) - \( \frac{e^{2 x}}{e^{2 x}+5} = \frac{7}{10} \) has solutions \( x = \frac{1}{2} \ln\left(\frac{35}{3}\right) \)