Answer :
To evaluate the logarithmic expression [tex]\(\ln \sqrt{3}\)[/tex], let's break it down step by step.
1. Understand the Expression:
We need to evaluate [tex]\(\ln \sqrt{3}\)[/tex]. Here, [tex]\(\ln\)[/tex] denotes the natural logarithm, which is the logarithm with base [tex]\(e\)[/tex], where [tex]\(e\)[/tex] is approximately equal to 2.71828.
2. Simplify the Argument:
The expression inside the logarithm is [tex]\(\sqrt{3}\)[/tex], which means the square root of 3.
3. Property of Logarithms:
Recall the property of logarithms that states [tex]\(\ln(\sqrt{a}) = \frac{1}{2} \ln(a)\)[/tex]. In this case, [tex]\(a = 3\)[/tex].
Therefore,
[tex]\[ \ln(\sqrt{3}) = \frac{1}{2} \ln(3) \][/tex]
4. Using the Property:
Now we just need the value of [tex]\(\ln(3)\)[/tex]. Using a calculator, we can determine that:
[tex]\[ \ln(3) \approx 1.0986 \][/tex]
5. Calculation:
Substitute the approximate value of [tex]\(\ln(3)\)[/tex] into our earlier equation:
[tex]\[ \ln(\sqrt{3}) = \frac{1}{2} \ln(3) \approx \frac{1}{2} \times 1.0986 \approx 0.5493 \][/tex]
Hence, the value of the logarithmic expression [tex]\(\ln \sqrt{3}\)[/tex] is approximately 0.549.
So the answer is:
[tex]\[ 0.549 \][/tex]
1. Understand the Expression:
We need to evaluate [tex]\(\ln \sqrt{3}\)[/tex]. Here, [tex]\(\ln\)[/tex] denotes the natural logarithm, which is the logarithm with base [tex]\(e\)[/tex], where [tex]\(e\)[/tex] is approximately equal to 2.71828.
2. Simplify the Argument:
The expression inside the logarithm is [tex]\(\sqrt{3}\)[/tex], which means the square root of 3.
3. Property of Logarithms:
Recall the property of logarithms that states [tex]\(\ln(\sqrt{a}) = \frac{1}{2} \ln(a)\)[/tex]. In this case, [tex]\(a = 3\)[/tex].
Therefore,
[tex]\[ \ln(\sqrt{3}) = \frac{1}{2} \ln(3) \][/tex]
4. Using the Property:
Now we just need the value of [tex]\(\ln(3)\)[/tex]. Using a calculator, we can determine that:
[tex]\[ \ln(3) \approx 1.0986 \][/tex]
5. Calculation:
Substitute the approximate value of [tex]\(\ln(3)\)[/tex] into our earlier equation:
[tex]\[ \ln(\sqrt{3}) = \frac{1}{2} \ln(3) \approx \frac{1}{2} \times 1.0986 \approx 0.5493 \][/tex]
Hence, the value of the logarithmic expression [tex]\(\ln \sqrt{3}\)[/tex] is approximately 0.549.
So the answer is:
[tex]\[ 0.549 \][/tex]
