\( \begin{array}{ll}\boldsymbol{t} \text { Check } & \text { Without using a calculator, give the value of } \log 10^{19.2} \\ \boldsymbol{t} \text { Check } & \text { Without using a calculator, give the value of } \ln e^{\sqrt{2}}\end{array} \)

The expression \( \log 10^{19.2} \) can be simplified using the properties of logarithms. Since \( \log a^b = b \cdot \log a \), we have \( \log 10^{19.2} = 19.2 \cdot \log 10 \). Given that \( \log 10 = 1 \), the final result is simply \( 19.2 \). For the second expression \( \ln e^{\sqrt{2}} \), we can use the same logarithmic property. Specifically, \( \ln a^b = b \cdot \ln a \) gives us \( \ln e^{\sqrt{2}} = \sqrt{2} \cdot \ln e \). Since \( \ln e = 1 \), the whole expression simplifies to \( \sqrt{2} \).