(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. \( 2^{x}=7 \) has solutions \( x=\square \) Hint: Webwork doesn't allow specifying the base of the logarithm, but you can use: \( \log _{B}(C)=\log (C) / \log (B) \) \( 5 \cdot 2^{x}=7 \) has solutions \( x=\square \) \( 5 \cdot 2^{x}+10=17 \) has solutions \( x=\square \) \( e^{x}=7 \) has solutions \( x=\square \) \( 5 \cdot e^{x}=7 \) has solutions \( x=\square \) \( 5 \cdot e^{x}+10=17 \) has solutions \( x=\square \)

To solve each of the equations, we will use logarithms where necessary. Let's go through each equation one by one. 1. **Equation:** \( 2^{x} = 7 \) Taking the logarithm of both sides: \[ x = \log_2(7) = \frac{\log(7)}{\log(2)} \] **Solution:** \( x = \frac{\log(7)}{\log(2)} \) 2. **Equation:** \( 5 \cdot 2^{x} = 7 \) Dividing both sides by 5: \[ 2^{x} = \frac{7}{5} \] Taking the logarithm of both sides: \[ x = \log_2\left(\frac{7}{5}\right) = \frac{\log\left(\frac{7}{5}\right)}{\log(2)} \] **Solution:** \( x = \frac{\log\left(\frac{7}{5}\right)}{\log(2)} \) 3. **Equation:** \( 5 \cdot 2^{x} + 10 = 17 \) Subtracting 10 from both sides: \[ 5 \cdot 2^{x} = 7 \] Dividing both sides by 5: \[ 2^{x} = \frac{7}{5} \] This is the same as the previous equation, so: \[ x = \log_2\left(\frac{7}{5}\right) = \frac{\log\left(\frac{7}{5}\right)}{\log(2)} \] **Solution:** \( x = \frac{\log\left(\frac{7}{5}\right)}{\log(2)} \) 4. **Equation:** \( e^{x} = 7 \) Taking the natural logarithm of both sides: \[ x = \ln(7) \] **Solution:** \( x = \ln(7) \) 5. **Equation:** \( 5 \cdot e^{x} = 7 \) Dividing both sides by 5: \[ e^{x} = \frac{7}{5} \] Taking the natural logarithm of both sides: \[ x = \ln\left(\frac{7}{5}\right) \] **Solution:** \( x = \ln\left(\frac{7}{5}\right) \) 6. **Equation:** \( 5 \cdot e^{x} + 10 = 17 \) Subtracting 10 from both sides: \[ 5 \cdot e^{x} = 7 \] Dividing both sides by 5: \[ e^{x} = \frac{7}{5} \] This is the same as the previous equation, so: \[ x = \ln\left(\frac{7}{5}\right) \] **Solution:** \( x = \ln\left(\frac{7}{5}\right) \) ### Summary of Solutions: - \( 2^{x} = 7 \) has solutions \( x = \frac{\log(7)}{\log(2)} \) - \( 5 \cdot 2^{x} = 7 \) has solutions \( x = \frac{\log\left(\frac{7}{5}\right)}{\log(2)} \) - \( 5 \cdot 2^{x} + 10 = 17 \) has solutions \( x = \frac{\log\left(\frac{7}{5}\right)}{\log(2)} \) - \( e^{x} = 7 \) has solutions \( x = \ln(7) \) - \( 5 \cdot e^{x} = 7 \) has solutions \( x = \ln\left(\frac{7}{5}\right) \) - \( 5 \cdot e^{x} + 10 = 17 \) has solutions \( x = \ln\left(\frac{7}{5}\right) \)