To simplify \( \log_{15} \log_{5} \), we can use the change of base formula for logarithms.
Using the change of base formula, we have:
\[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \]
For \( \log_{15} \) and \( \log_{5} \), we can use any base (commonly base 10 or base \( e \)). Choosing base \( 10 \) for simplicity, we can write:
\[ \log_{15} = \frac{\log_{10}}{\log_{10} 15} \] \[ \log_{5} = \frac{\log_{10}}{\log_{10} 5} \]
Now, substituting \( \log_{5} \) into \( \log_{15} \):
\[ \log_{15} \log_{5} = \log_{15} \left(\frac{\log_{10}}{\log_{10} 5}\right) \]
However, since we need to simplify \( \log_{15} \log_{5} \) in more manageable terms:
First, let's recognize \( \log_{15} \) itself:
\[ \log_{15} = \log_{(3 \cdot 5)} = \log_{3} + \log_{5} \]
Thus, the original expression is just \( \log_{5} 15 \):
This is not directly in the choices given but since no additional context is given, we can also conclude that:
If we take the values to their simplest expressions in terms of single variable logarithms, we can notice:
\[ \log_{15} \log_{5} = \frac{\log_{5} 15}{\log_{5}} = \log_{5} 15 \]
None of the direct transformation yields a direct option given seems like it doesn't apply clearly to existing options.
Nonetheless:
So your correct substitution is \( \log_{5} 15\) represents your final equation.
However, considering the options:
- Log base 3 might be a misinterpret; there could be typographical absence as nothing precisely matches the transformation in logarithms.
In summary, double-check the mentioned options since \( \log_{15} \) does not simplify into one of the choices without additional context on variable base conditions that influence the logarithmic gradient across given transformations.
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