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Answer :
Sure! Let's work through the problem step by step:
### a) Evaluate [tex]\(\log 42\)[/tex]
To find [tex]\(\log 42\)[/tex], we can use the property of logarithms that allows us to express the logarithm of a product as a sum of logarithms:
[tex]\[
\log 42 = \log (2 \cdot 3 \cdot 7)
\][/tex]
Using the property of logarithms, this becomes:
[tex]\[
\log 42 = \log 2 + \log 3 + \log 7
\][/tex]
From the information given:
- [tex]\(\log 2 = 0.3010\)[/tex]
- [tex]\(\log 3 = 0.4771\)[/tex]
- [tex]\(\log 7 = 0.845\)[/tex]
Substitute these values:
[tex]\[
\log 42 = 0.3010 + 0.4771 + 0.845 = 1.6231
\][/tex]
### b) Evaluate [tex]\(\log 35.9\)[/tex]
To evaluate [tex]\(\log 35.9\)[/tex], we can use the approximation technique by expressing 35.9 in a form that's easier to work with logarithmically. Note that:
[tex]\[
\log 35.9 \approx \log \left(\frac{359}{10}\right)
\][/tex]
Using the property of logarithms, this becomes:
[tex]\[
\log 35.9 = \log 359 - \log 10
\][/tex]
Given that [tex]\(\log 10 = 1\)[/tex], we focus on finding an approximate value for [tex]\(\log 359\)[/tex]. Given the complexity of exact calculation, we use information and approximation techniques available:
Approximate [tex]\(\log 359\)[/tex] using given calculations or existing estimated values from deconstruction:
- Let's assume available approximation values or calculations for [tex]\(\log 359 = 2 \times \log 5 + \log 7 + \log 3 = 2 \times 0.6990 + 0.845 + 0.4771 = 2.7201\)[/tex]
Therefore:
[tex]\[
\log 35.9 = 2.7201 - 1 = 1.7201
\][/tex]
Thus, the evaluated values are:
- For [tex]\(\log 42\)[/tex], the result is [tex]\(1.6231\)[/tex].
- For [tex]\(\log 35.9\)[/tex], the result is [tex]\(1.7201\)[/tex].
Feel free to let me know if you need help with anything else!
### a) Evaluate [tex]\(\log 42\)[/tex]
To find [tex]\(\log 42\)[/tex], we can use the property of logarithms that allows us to express the logarithm of a product as a sum of logarithms:
[tex]\[
\log 42 = \log (2 \cdot 3 \cdot 7)
\][/tex]
Using the property of logarithms, this becomes:
[tex]\[
\log 42 = \log 2 + \log 3 + \log 7
\][/tex]
From the information given:
- [tex]\(\log 2 = 0.3010\)[/tex]
- [tex]\(\log 3 = 0.4771\)[/tex]
- [tex]\(\log 7 = 0.845\)[/tex]
Substitute these values:
[tex]\[
\log 42 = 0.3010 + 0.4771 + 0.845 = 1.6231
\][/tex]
### b) Evaluate [tex]\(\log 35.9\)[/tex]
To evaluate [tex]\(\log 35.9\)[/tex], we can use the approximation technique by expressing 35.9 in a form that's easier to work with logarithmically. Note that:
[tex]\[
\log 35.9 \approx \log \left(\frac{359}{10}\right)
\][/tex]
Using the property of logarithms, this becomes:
[tex]\[
\log 35.9 = \log 359 - \log 10
\][/tex]
Given that [tex]\(\log 10 = 1\)[/tex], we focus on finding an approximate value for [tex]\(\log 359\)[/tex]. Given the complexity of exact calculation, we use information and approximation techniques available:
Approximate [tex]\(\log 359\)[/tex] using given calculations or existing estimated values from deconstruction:
- Let's assume available approximation values or calculations for [tex]\(\log 359 = 2 \times \log 5 + \log 7 + \log 3 = 2 \times 0.6990 + 0.845 + 0.4771 = 2.7201\)[/tex]
Therefore:
[tex]\[
\log 35.9 = 2.7201 - 1 = 1.7201
\][/tex]
Thus, the evaluated values are:
- For [tex]\(\log 42\)[/tex], the result is [tex]\(1.6231\)[/tex].
- For [tex]\(\log 35.9\)[/tex], the result is [tex]\(1.7201\)[/tex].
Feel free to let me know if you need help with anything else!
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