Evaluate the following expressions. In part 3 below your answer needs to be a decimal rounded to three places. The rest of your answers must be exact and in simplest form. \( \ln e^{21}= \) \( \ln (1)= \) \( \ln \left(e^{\sqrt{15}}\right)= \) \( \ln \left(\frac{1}{e^{3}}\right)= \)

Let's evaluate each expression step by step. 1. **Evaluate \( \ln e^{21} \)**: \[ \ln e^{21} = 21 \] (Using the property \( \ln e^x = x \)) 2. **Evaluate \( \ln (1) \)**: \[ \ln (1) = 0 \] (Since \( e^0 = 1 \)) 3. **Evaluate \( \ln \left(e^{\sqrt{15}}\right) \)**: \[ \ln \left(e^{\sqrt{15}}\right) = \sqrt{15} \] (Using the property \( \ln e^x = x \)) 4. **Evaluate \( \ln \left(\frac{1}{e^{3}}\right) \)**: \[ \ln \left(\frac{1}{e^{3}}\right) = \ln \left(e^{-3}\right) = -3 \] (Using the property \( \ln \left(\frac{1}{x}\right) = -\ln x \) and \( \ln e^x = x \)) Now, for part 3, we need to provide the answer as a decimal rounded to three places: \[ \sqrt{15} \approx 3.873 \] So, the final answers are: 1. \( \ln e^{21} = 21 \) 2. \( \ln (1) = 0 \) 3. \( \ln \left(e^{\sqrt{15}}\right) \approx 3.873 \) 4. \( \ln \left(\frac{1}{e^{3}}\right) = -3 \)