Answer :
Sure, let's solve each part of the problem step-by-step.
### Part 1: Calculating [tex]\(\log_{10}(20.6)\)[/tex]
1. Understand what [tex]\(\log_{10}(20.6)\)[/tex] means: The [tex]\(\log_{10}\)[/tex] function represents the logarithm with base 10. It answers the question: "To what power must 10 be raised, to yield 20.6?".
2. Find the value: Using a calculator, we find the value of [tex]\(\log_{10}(20.6)\)[/tex].
3. Round the result to the nearest thousandth: The computed value is approximately 1.314 when rounded to three decimal places.
So, [tex]\(\log_{10}(20.6) \approx 1.314\)[/tex].
### Part 2: Calculating [tex]\(\ln(\sqrt{3})\)[/tex]
1. Understand what [tex]\(\ln(\sqrt{3})\)[/tex] means: [tex]\(\ln\)[/tex] represents the natural logarithm, which is the logarithm with base [tex]\(e\)[/tex] (where [tex]\(e \approx 2.718\)[/tex]). The [tex]\(\sqrt{3}\)[/tex] indicates the square root of 3.
2. Calculate the square root of 3: First, find the value of [tex]\(\sqrt{3}\)[/tex].
3. Find the natural logarithm of [tex]\(\sqrt{3}\)[/tex]: Using a calculator, find the value of [tex]\(\ln(\sqrt{3})\)[/tex].
4. Round the result to the nearest thousandth: The calculated value is approximately 0.549 when rounded to three decimal places.
So, [tex]\(\ln(\sqrt{3}) \approx 0.549\)[/tex].
### Final Answer:
[tex]\[ \begin{array}{l} \log_{10}(20.6) \approx 1.314 \\ \ln(\sqrt{3}) \approx 0.549 \\ \end{array} \][/tex]
### Part 1: Calculating [tex]\(\log_{10}(20.6)\)[/tex]
1. Understand what [tex]\(\log_{10}(20.6)\)[/tex] means: The [tex]\(\log_{10}\)[/tex] function represents the logarithm with base 10. It answers the question: "To what power must 10 be raised, to yield 20.6?".
2. Find the value: Using a calculator, we find the value of [tex]\(\log_{10}(20.6)\)[/tex].
3. Round the result to the nearest thousandth: The computed value is approximately 1.314 when rounded to three decimal places.
So, [tex]\(\log_{10}(20.6) \approx 1.314\)[/tex].
### Part 2: Calculating [tex]\(\ln(\sqrt{3})\)[/tex]
1. Understand what [tex]\(\ln(\sqrt{3})\)[/tex] means: [tex]\(\ln\)[/tex] represents the natural logarithm, which is the logarithm with base [tex]\(e\)[/tex] (where [tex]\(e \approx 2.718\)[/tex]). The [tex]\(\sqrt{3}\)[/tex] indicates the square root of 3.
2. Calculate the square root of 3: First, find the value of [tex]\(\sqrt{3}\)[/tex].
3. Find the natural logarithm of [tex]\(\sqrt{3}\)[/tex]: Using a calculator, find the value of [tex]\(\ln(\sqrt{3})\)[/tex].
4. Round the result to the nearest thousandth: The calculated value is approximately 0.549 when rounded to three decimal places.
So, [tex]\(\ln(\sqrt{3}) \approx 0.549\)[/tex].
### Final Answer:
[tex]\[ \begin{array}{l} \log_{10}(20.6) \approx 1.314 \\ \ln(\sqrt{3}) \approx 0.549 \\ \end{array} \][/tex]
