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Answer :
To evaluate the expression [tex]\(\log_5 \frac{1}{25}\)[/tex] without using a calculator, follow these steps:
1. Rewrite the fraction: Understand that [tex]\(\frac{1}{25}\)[/tex] is the same as [tex]\(25^{-1}\)[/tex]. Therefore, [tex]\(\log_5 \frac{1}{25}\)[/tex] can be rewritten as [tex]\(\log_5 (25^{-1})\)[/tex].
2. Express 25 as a power of 5: Since 25 is [tex]\(5^2\)[/tex], we can substitute this into the expression. Thus, we have [tex]\(\log_5 ((5^2)^{-1})\)[/tex].
3. Apply the power of a power property: According to the rules of exponents, [tex]\((a^b)^c = a^{b \cdot c}\)[/tex]. Applying this property, [tex]\((5^2)^{-1} = 5^{-2}\)[/tex].
4. Use the power rule of logarithms: The power rule states that [tex]\(\log_b (a^c) = c \cdot \log_b (a)\)[/tex]. Applying this rule, [tex]\(\log_5 (5^{-2}) = -2 \cdot \log_5 (5)\)[/tex].
5. Evaluate [tex]\(\log_5 (5)\)[/tex]: Since any logarithm [tex]\(\log_b (b)\)[/tex] is equal to 1, we have [tex]\(\log_5 (5) = 1\)[/tex].
6. Calculate the final expression: Substitute back into the expression, [tex]\(-2 \cdot \log_5 (5) = -2 \cdot 1 = -2\)[/tex].
The value of [tex]\(\log_5 \frac{1}{25}\)[/tex] is [tex]\(-2\)[/tex].
1. Rewrite the fraction: Understand that [tex]\(\frac{1}{25}\)[/tex] is the same as [tex]\(25^{-1}\)[/tex]. Therefore, [tex]\(\log_5 \frac{1}{25}\)[/tex] can be rewritten as [tex]\(\log_5 (25^{-1})\)[/tex].
2. Express 25 as a power of 5: Since 25 is [tex]\(5^2\)[/tex], we can substitute this into the expression. Thus, we have [tex]\(\log_5 ((5^2)^{-1})\)[/tex].
3. Apply the power of a power property: According to the rules of exponents, [tex]\((a^b)^c = a^{b \cdot c}\)[/tex]. Applying this property, [tex]\((5^2)^{-1} = 5^{-2}\)[/tex].
4. Use the power rule of logarithms: The power rule states that [tex]\(\log_b (a^c) = c \cdot \log_b (a)\)[/tex]. Applying this rule, [tex]\(\log_5 (5^{-2}) = -2 \cdot \log_5 (5)\)[/tex].
5. Evaluate [tex]\(\log_5 (5)\)[/tex]: Since any logarithm [tex]\(\log_b (b)\)[/tex] is equal to 1, we have [tex]\(\log_5 (5) = 1\)[/tex].
6. Calculate the final expression: Substitute back into the expression, [tex]\(-2 \cdot \log_5 (5) = -2 \cdot 1 = -2\)[/tex].
The value of [tex]\(\log_5 \frac{1}{25}\)[/tex] is [tex]\(-2\)[/tex].
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