To address the given problem, we can use the properties of logarithms. We are given the expression:
[tex]\[
\log_5(25b)
\][/tex]
We need to express this logarithm as a sum of logarithms. To do that, we use the product property of logarithms, which states:
[tex]\[
\log_b(mn) = \log_b(m) + \log_b(n)
\][/tex]
Applying this property, we can split the logarithm of the product [tex]\(25b\)[/tex] into the sum of two logarithms:
[tex]\[
\log_5(25b) = \log_5(25) + \log_5(b)
\][/tex]
Next, we need to simplify [tex]\(\log_5(25)\)[/tex]. Notice that 25 can be expressed as [tex]\(5^2\)[/tex]. Therefore, we have:
[tex]\[
\log_5(25) = \log_5(5^2)
\][/tex]
Using another property of logarithms, which is the power rule, [tex]\( \log_b(m^n) = n \cdot \log_b(m) \)[/tex], we can simplify [tex]\(\log_5(5^2)\)[/tex]:
[tex]\[
\log_5(5^2) = 2 \cdot \log_5(5)
\][/tex]
Since the logarithm of a base to itself is always 1, i.e., [tex]\( \log_5(5) = 1 \)[/tex], we get:
[tex]\[
\log_5(5^2) = 2 \cdot 1 = 2
\][/tex]
Putting this all together, we have:
[tex]\[
\log_5(25) + \log_5(b) = 2 + \log_5(b)
\][/tex]
Thus, the original expression simplifies to:
[tex]\[
\log_5(25b) = 2 + \log_5(b)
\][/tex]
So, the logarithm [tex]\(\log_5(25b)\)[/tex] can be written as a sum and simplified to:
[tex]\[
\boxed{2 + \log_5(b)}
\][/tex]
