Question

Simplify log15log5
log
⁡
15
log
⁡
5
.(1 point)
Responses

log3
log
⁡
3
logarithm 3

3
3

log515
log
5
⁡
15
logarithm subscript 5 baseline 15

log155

Answer 1

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.