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Answer :
To convert the equation [tex]\(3125^{3/5}=125\)[/tex] into logarithmic form, we need to use the logarithm properties that relate exponents to logarithms. The relationship can be written as:
[tex]\[a^b = c \quad \text{is equivalent to} \quad \log_a(c) = b\][/tex]
For the given equation:
[tex]\[3125^{3/5} = 125\][/tex]
Here, the base [tex]\(a\)[/tex] is [tex]\(3125\)[/tex], the exponent [tex]\(b\)[/tex] is [tex]\(\frac{3}{5}\)[/tex], and the result [tex]\(c\)[/tex] is [tex]\(125\)[/tex].
Using the equivalent logarithmic form, we can write:
[tex]\[\log_{3125}(125) = \frac{3}{5}\][/tex]
Now, let's match this with the options given:
(A) [tex]\(\log_{3125} 125=\frac{3}{5}\)[/tex]
(B) [tex]\(\log_{3 / 5} 125=3125\)[/tex]
(C) [tex]\(5 \log_3 125=3125\)[/tex]
(D) [tex]\(\log_{125} 3125=\frac{5}{3}\)[/tex]
We see that option (A) is exactly what we derived:
[tex]\[\log_{3125} 125 = \frac{3}{5}\][/tex]
Therefore, the correct answer is:
(A) [tex]\(\log_{3125} 125 = \frac{3}{5}\)[/tex]
[tex]\[a^b = c \quad \text{is equivalent to} \quad \log_a(c) = b\][/tex]
For the given equation:
[tex]\[3125^{3/5} = 125\][/tex]
Here, the base [tex]\(a\)[/tex] is [tex]\(3125\)[/tex], the exponent [tex]\(b\)[/tex] is [tex]\(\frac{3}{5}\)[/tex], and the result [tex]\(c\)[/tex] is [tex]\(125\)[/tex].
Using the equivalent logarithmic form, we can write:
[tex]\[\log_{3125}(125) = \frac{3}{5}\][/tex]
Now, let's match this with the options given:
(A) [tex]\(\log_{3125} 125=\frac{3}{5}\)[/tex]
(B) [tex]\(\log_{3 / 5} 125=3125\)[/tex]
(C) [tex]\(5 \log_3 125=3125\)[/tex]
(D) [tex]\(\log_{125} 3125=\frac{5}{3}\)[/tex]
We see that option (A) is exactly what we derived:
[tex]\[\log_{3125} 125 = \frac{3}{5}\][/tex]
Therefore, the correct answer is:
(A) [tex]\(\log_{3125} 125 = \frac{3}{5}\)[/tex]
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