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Which equation is equivalent to [tex]\ell^m=n[/tex]?

A. [tex]\ell=\log_m n[/tex]

B. [tex]m=\log_n \ell[/tex]

C. [tex]m=1 \int^n[/tex]

D. [tex]\ell=\log_\pi m[/tex]

Sagot :

GinnyAnsweridn
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]
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