To find the value of [tex]\(\log_5(125) + \log_5\left(5^7\right)\)[/tex], let's break it down step by step.
1. Evaluate [tex]\(\log_5(125)\)[/tex]:
- We recognize that [tex]\(125\)[/tex] can be expressed as a power of [tex]\(5\)[/tex]: [tex]\(125 = 5^3\)[/tex].
- Using the property of logarithms [tex]\(\log_b(b^x) = x\)[/tex], we have:
[tex]\[
\log_5(125) = \log_5(5^3) = 3.0000000000000004
\][/tex]
2. Evaluate [tex]\(\log_5\left(5^7\right)\)[/tex]:
- Here, [tex]\(5^7\)[/tex] is already expressed as a power of [tex]\(5\)[/tex].
- Again using the property of logarithms [tex]\(\log_b(b^x) = x\)[/tex], we get:
[tex]\[
\log_5(5^7) = 7.0
\][/tex]
3. Sum the logarithms:
- Now, add the two logarithmic results together:
[tex]\[
\log_5(125) + \log_5(5^7) = 3.0000000000000004 + 7.0 = 10.0
\][/tex]
Thus, the value of [tex]\(\log_5(125) + \log_5\left(5^7\right)\)[/tex] is [tex]\(10.0\)[/tex].
