Answer :
Sure! Let's solve the problem step by step.
We are given the equation [tex]\(5^5 = 3125\)[/tex] and need to identify which logarithmic expression correctly represents this equation from the options given.
By the definition of logarithms, if [tex]\(a^b = c\)[/tex], then [tex]\(\log_a c = b\)[/tex]. This means that if you have a number raised to a power that equals another number, you can express this relationship using a logarithm.
In the given equation [tex]\(5^5 = 3125\)[/tex]:
- [tex]\(a\)[/tex] is 5 (the base),
- [tex]\(b\)[/tex] is 5 (the exponent),
- [tex]\(c\)[/tex] is 3125 (the result).
According to the logarithm property, we have:
[tex]\[
\log_5 3125 = 5
\][/tex]
Let's compare this with the options:
A. [tex]\(\log_5 5 = 3125\)[/tex] - This is incorrect because it suggests the result is 3125.
B. [tex]\(\log_5 3125 = 5\)[/tex] - This is correct as it matches our derived expression.
C. [tex]\(\log_{3125} 5 = 5\)[/tex] - This is incorrect because it suggests the base is 3125 and the exponent is 5.
D. [tex]\(\log_5 5 = 3125\)[/tex] - This is identical to A and is not correct.
Thus, the correct answer is B: [tex]\(\log_5 3125 = 5\)[/tex].
We are given the equation [tex]\(5^5 = 3125\)[/tex] and need to identify which logarithmic expression correctly represents this equation from the options given.
By the definition of logarithms, if [tex]\(a^b = c\)[/tex], then [tex]\(\log_a c = b\)[/tex]. This means that if you have a number raised to a power that equals another number, you can express this relationship using a logarithm.
In the given equation [tex]\(5^5 = 3125\)[/tex]:
- [tex]\(a\)[/tex] is 5 (the base),
- [tex]\(b\)[/tex] is 5 (the exponent),
- [tex]\(c\)[/tex] is 3125 (the result).
According to the logarithm property, we have:
[tex]\[
\log_5 3125 = 5
\][/tex]
Let's compare this with the options:
A. [tex]\(\log_5 5 = 3125\)[/tex] - This is incorrect because it suggests the result is 3125.
B. [tex]\(\log_5 3125 = 5\)[/tex] - This is correct as it matches our derived expression.
C. [tex]\(\log_{3125} 5 = 5\)[/tex] - This is incorrect because it suggests the base is 3125 and the exponent is 5.
D. [tex]\(\log_5 5 = 3125\)[/tex] - This is identical to A and is not correct.
Thus, the correct answer is B: [tex]\(\log_5 3125 = 5\)[/tex].
