To find the equation that is equivalent to [tex]\(\ell^m = n\)[/tex], we need to use logarithmic properties to manipulate and transform the initial equation.
1. Start with the given equation:
[tex]\[\ell^m = n\][/tex]
2. To solve for [tex]\(m\)[/tex], take the logarithm of both sides. For simplicity, we'll use the base [tex]\(n\)[/tex] logarithm ([tex]\(\log_n\)[/tex]):
[tex]\[\log_n(\ell^m) = \log_n(n)\][/tex]
3. Applying the logarithmic property [tex]\(\log_b(a^c) = c \log_b(a)\)[/tex], we get:
[tex]\[m \log_n(\ell) = \log_n(n)\][/tex]
4. Since [tex]\(\log_n(n) = 1\)[/tex] (because any number to the base of itself is 1), the equation simplifies to:
[tex]\[m \log_n(\ell) = 1\][/tex]
5. Now, isolate [tex]\(m\)[/tex] by dividing both sides by [tex]\(\log_n(\ell)\)[/tex]:
[tex]\[m = \frac{1}{\log_n(\ell)}\][/tex]
6. Recognize that [tex]\(\frac{1}{\log_n(\ell)}\)[/tex] could be rewritten using the change of base formula, but even simpler, we already understand that by properties of logarithms:
[tex]\[m = \log_\ell (n)\][/tex]
So the equation [tex]\(m = \log_n (\ell)\)[/tex] matches perfectly as the equivalent to the original equation [tex]\(\ell^m = n\)[/tex].
Hence, the correct option is:
[tex]\[m = \log_n(\ell)\][/tex]
Therefore, the answer is:
[tex]\[1\][/tex]
