To solve for [tex]\(\log 1250\)[/tex] in terms of [tex]\(b\)[/tex], where [tex]\(b = \log 5\)[/tex], we can use properties of logarithms and the provided logarithm values.
First, let's break down 1250 using its prime factors:
[tex]\[
1250 = 5^3 \times 2^1 \times 5^1 = 5^4 \times 2
\][/tex]
Given the properties of logarithms:
1. [tex]\(\log(ab) = \log a + \log b\)[/tex]
2. [tex]\(\log(a^b) = b \log a\)[/tex]
Let's apply these properties step-by-step:
1. Express [tex]\(\log 1250\)[/tex] in terms of [tex]\(\log (5^4 \times 2)\)[/tex]:
[tex]\[
\log 1250 = \log (5^4 \times 2)
\][/tex]
2. Apply the logarithm property [tex]\(\log(ab) = \log a + \log b\)[/tex] to split the logarithms:
[tex]\[
\log (5^4 \times 2) = \log 5^4 + \log 2
\][/tex]
3. Apply the logarithm property [tex]\(\log(a^b) = b \log a\)[/tex] to [tex]\(\log 5^4\)[/tex]:
[tex]\[
\log 5^4 = 4 \log 5
\][/tex]
By substituting [tex]\(\log 5\)[/tex] with [tex]\(b\)[/tex], we get:
[tex]\[
4 \log 5 = 4b
\][/tex]
4. Now, we need the value of [tex]\(\log 2\)[/tex]. Given the solution, we have:
[tex]\[
\log 2 = 0.3010299956639812
\][/tex]
Putting it all together, [tex]\(\log 1250\)[/tex]:
[tex]\[
\log 1250 = 4b + \log 2
\][/tex]
Finally, substituting [tex]\(\log 2\)[/tex]:
[tex]\[
\log 1250 = 4b + 0.3010299956639812
\][/tex]
By evaluating the values of [tex]\(b = \log 5 = 0.6989700043360189\)[/tex] and using the provided logarithmic values:
[tex]\[
\log 1250 = 4 \times 0.6989700043360189 + 0.3010299956639812 = 3.0969100130080567
\][/tex]
Thus, [tex]\(\log 1250\)[/tex] in terms of [tex]\(b\)[/tex] is:
[tex]\[
\log 1250 = 4b + 0.3010299956639812 = 3.0969100130080567.
\][/tex]
This completes the detailed step-by-step solution.
