Certainly! Let's use the change-of-base theorem to rewrite the given logarithmic expression.
The change-of-base theorem states that for any positive numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (where [tex]\(a \neq 1\)[/tex] and [tex]\(b \neq 1\)[/tex]), the logarithm of [tex]\(b\)[/tex] with base [tex]\(a\)[/tex] can be rewritten as a ratio of logarithms with a new base [tex]\(c\)[/tex]:
[tex]\[
\log_a b = \frac{\log_c b}{\log_c a}
\][/tex]
In this problem, we are given [tex]\(\log_{\frac{1}{5}} 4\)[/tex] and we want to rewrite it using the change-of-base theorem.
Here, [tex]\(a = \frac{1}{5}\)[/tex] and [tex]\(b = 4\)[/tex]. We can choose any base [tex]\(c\)[/tex] for the logarithms; commonly used bases are [tex]\(10\)[/tex] (common logarithm) or [tex]\(e\)[/tex] (natural logarithm). For this solution, we will choose the natural logarithm (base [tex]\(e\)[/tex]).
Now, applying the change-of-base theorem:
[tex]\[
\log_{\frac{1}{5}} 4 = \frac{\log_e 4}{\log_e \left(\frac{1}{5}\right)}
\][/tex]
In general terms, it can also be written with common logarithms (base 10):
[tex]\[
\log_{\frac{1}{5}} 4 = \frac{\log 4}{\log \left(\frac{1}{5}\right)}
\][/tex]
Both forms are valid since the choice of [tex]\(e\)[/tex] or [tex]\(10\)[/tex] doesn't affect the outcome as long as the same base is used in both the numerator and the denominator.
So, the rewritten expression is:
[tex]\[
\log_{\frac{1}{5}} 4 = \frac{\log 4}{\log \left(\frac{1}{5}\right)}
\][/tex]
And that's the result from applying the change-of-base theorem to the given logarithmic expression.
