Sure, let's break down and simplify the given logarithmic expression:
[tex]\[ \log_5\left(\frac{1}{5} m m^2\right) \][/tex]
1. Combine the terms inside the logarithm:
[tex]\[ \frac{1}{5} m m^2 = \frac{1}{5} m^3 \][/tex]
Thus, the expression becomes:
[tex]\[ \log_5\left(\frac{1}{5} m^3\right) \][/tex]
2. Use properties of logarithms to separate the terms:
[tex]\[ \log_5(a \cdot b) = \log_5(a) + \log_5(b) \][/tex]
Here, [tex]\( a = \frac{1}{5} \)[/tex] and [tex]\( b = m^3 \)[/tex]:
[tex]\[ \log_5\left(\frac{1}{5} m^3\right) = \log_5\left(\frac{1}{5}\right) + \log_5(m^3) \][/tex]
3. Simplify each term individually:
- For the first term, [tex]\(\log_5\left(\frac{1}{5}\right)\)[/tex]:
[tex]\[ \frac{1}{5} = 5^{-1} \][/tex]
Thus:
[tex]\[ \log_5\left(5^{-1}\right) = -1 \cdot \log_5(5) \][/tex]
Since [tex]\(\log_5(5) = 1\)[/tex]:
[tex]\[ \log_5\left(5^{-1}\right) = -1 \][/tex]
- For the second term, [tex]\(\log_5(m^3)\)[/tex]:
[tex]\[ \log_5(m^3) = 3 \cdot \log_5(m) \][/tex]
4. Combine the simplified terms:
[tex]\[ \log_5\left(\frac{1}{5} m^3\right) = -1 + 3 \cdot \log_5(m) \][/tex]
Therefore, the simplified expression is:
[tex]\[ \boxed{-1 + 3 \cdot \log_5(m)} \][/tex]
