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Answer :
Sure! Let's go through the steps to convert the logarithmic equation to its exponential form.
We are given the logarithmic equation:
[tex]\[ \log _5 \frac{1}{25} = -2 \][/tex]
To convert this logarithmic equation to its exponential form, we can use the following property of logarithms:
[tex]\[ \log_b(a) = c \Longleftrightarrow b^c = a \][/tex]
In our case:
- [tex]\( b = 5 \)[/tex] (the base of the logarithm)
- [tex]\( a = \frac{1}{25} \)[/tex] (the number we are taking the logarithm of)
- [tex]\( c = -2 \)[/tex] (the result of the logarithm)
Using these values, we can rewrite the logarithmic equation [tex]\(\log _5 \frac{1}{25} = -2\)[/tex] in its exponential form as:
[tex]\[ 5^{-2} = \frac{1}{25} \][/tex]
Now we can match this result with the given multiple-choice options:
(A) [tex]\( 5^{-2} = \frac{1}{25} \)[/tex]
(B) [tex]\( 2^5 = -\frac{1}{25} \)[/tex]
(C) [tex]\( 5^{-2} = 25 \)[/tex]
(D) [tex]\( 2^5 = -25 \)[/tex]
The correct exponential form is:
[tex]\[ 5^{-2} = \frac{1}{25} \][/tex]
So, the correct answer is:
(A) [tex]\( 5^{-2} = \frac{1}{25} \)[/tex]
We are given the logarithmic equation:
[tex]\[ \log _5 \frac{1}{25} = -2 \][/tex]
To convert this logarithmic equation to its exponential form, we can use the following property of logarithms:
[tex]\[ \log_b(a) = c \Longleftrightarrow b^c = a \][/tex]
In our case:
- [tex]\( b = 5 \)[/tex] (the base of the logarithm)
- [tex]\( a = \frac{1}{25} \)[/tex] (the number we are taking the logarithm of)
- [tex]\( c = -2 \)[/tex] (the result of the logarithm)
Using these values, we can rewrite the logarithmic equation [tex]\(\log _5 \frac{1}{25} = -2\)[/tex] in its exponential form as:
[tex]\[ 5^{-2} = \frac{1}{25} \][/tex]
Now we can match this result with the given multiple-choice options:
(A) [tex]\( 5^{-2} = \frac{1}{25} \)[/tex]
(B) [tex]\( 2^5 = -\frac{1}{25} \)[/tex]
(C) [tex]\( 5^{-2} = 25 \)[/tex]
(D) [tex]\( 2^5 = -25 \)[/tex]
The correct exponential form is:
[tex]\[ 5^{-2} = \frac{1}{25} \][/tex]
So, the correct answer is:
(A) [tex]\( 5^{-2} = \frac{1}{25} \)[/tex]
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