Question
\( \begin{array}{ll}\boldsymbol{t} \text { Check } & \text { Without using a calculator, give the value of } \log 10^{19.2} \\ \boldsymbol{t} \text { Check } & \text { Without using a calculator, give the value of } \ln e^{\sqrt{2}}\end{array} \)
Ask by Todd Robbins. in the United States
Feb 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
1) log(10^(19.2)) = 19.2
2) ln(e^(√2)) = √2
Solution
1) The value of log(10^(19.2)) is 19.2.
Explanation: For any base b (here, b = 10) and exponent x, log_b(b^x) = x.
2) The value of ln(e^(√2)) is √2.
Explanation: Since the natural logarithm ln is base e, for any exponent x, ln(e^x) = x.
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Simplify this solution Extra Insights
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 \( \sqrt{2} \).
