To solve for [tex]\(\log 1250\)[/tex] in terms of [tex]\(\log 5 = b\)[/tex], we proceed through the following steps:
1. Express 1250 in terms of its prime factors:
[tex]\[
1250 = 125 \times 10 = 5^3 \times (2 \times 5) = 5^3 \times 2 \times 5^2 = 5^5 \times 2
\][/tex]
2. Utilize the properties of logarithms:
One of the key properties of logarithms is that [tex]\(\log(ab) = \log a + \log b\)[/tex] and [tex]\(\log(a^b) = b \log a\)[/tex]. Applying these properties, we can rewrite [tex]\(\log 1250\)[/tex]:
[tex]\[
\log 1250 = \log(5^5 \times 2) = \log(5^5) + \log 2
\][/tex]
Using the power property of logarithms:
[tex]\[
\log(5^5) = 5 \log 5
\][/tex]
3. Substitute our given [tex]\( \log 5 = b \)[/tex]:
Let [tex]\( \log 5 = b \)[/tex], thus, we have:
[tex]\[
\log(5^5) = 5b
\][/tex]
Now, substitute it back into the equation for [tex]\(\log 1250\)[/tex]:
[tex]\[
\log 1250 = 5 \log 5 + \log 2 = 5b + \log 2
\][/tex]
4. Determine the value for [tex]\(\log 2\)[/tex]:
In mathematical terms:
[tex]\[
\log 2 \approx 0.6931471805599453
\][/tex]
5. Combining the values:
Given that [tex]\(b = \log 5 \approx 1.6094379124341003\)[/tex]:
[tex]\[
5b = 5 \times 1.6094379124341003 = 8.0471895621705015
\][/tex]
Combining these, the final expression for [tex]\(\log 1250\)[/tex] is:
[tex]\[
\log 1250 = 5b + \log 2 = 8.0471895621705015 + 0.6931471805599453 \approx 8.740336742730447
\][/tex]
Thus, in terms of [tex]\(b\)[/tex], [tex]\(\log 1250 = 5b + \log 2\)[/tex], and numerically:
[tex]\[
\log 1250 \approx 8.740336742730447
\][/tex]
