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Answer :
To evaluate the expression [tex]\(\log_5 \frac{1}{25}\)[/tex] without using a calculator, we can follow these steps:
1. Understand the Expression:
[tex]\(\log_5 \frac{1}{25}\)[/tex] asks us to find the exponent to which 5 must be raised to get [tex]\(\frac{1}{25}\)[/tex].
2. Rewrite the Fraction as a Power:
We recognize that [tex]\(\frac{1}{25}\)[/tex] can be rewritten using a negative exponent:
[tex]\[
\frac{1}{25} = 25^{-1}
\][/tex]
3. Apply the Change of Base:
Using the property of logarithms that [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex], we rewrite the expression:
[tex]\[
\log_5 \left(25^{-1}\right) = -1 \cdot \log_5(25)
\][/tex]
4. Simplify [tex]\(\log_5(25)\)[/tex]:
Notice that 25 can be rewritten as [tex]\(5^2\)[/tex], so:
[tex]\[
\log_5(25) = \log_5(5^2)
\][/tex]
5. Use the Power Rule Again:
Applying the power rule once more:
[tex]\[
\log_5(5^2) = 2 \cdot \log_5(5)
\][/tex]
Since [tex]\(\log_5(5) = 1\)[/tex] (because anything log_b(b) is 1):
[tex]\[
2 \cdot \log_5(5) = 2 \cdot 1 = 2
\][/tex]
6. Substitute Back:
Now substitute back to our previous expression:
[tex]\[
-1 \cdot \log_5(25) = -1 \cdot 2 = -2
\][/tex]
Therefore, the value of [tex]\(\log_5 \frac{1}{25}\)[/tex] is [tex]\(-2\)[/tex].
1. Understand the Expression:
[tex]\(\log_5 \frac{1}{25}\)[/tex] asks us to find the exponent to which 5 must be raised to get [tex]\(\frac{1}{25}\)[/tex].
2. Rewrite the Fraction as a Power:
We recognize that [tex]\(\frac{1}{25}\)[/tex] can be rewritten using a negative exponent:
[tex]\[
\frac{1}{25} = 25^{-1}
\][/tex]
3. Apply the Change of Base:
Using the property of logarithms that [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex], we rewrite the expression:
[tex]\[
\log_5 \left(25^{-1}\right) = -1 \cdot \log_5(25)
\][/tex]
4. Simplify [tex]\(\log_5(25)\)[/tex]:
Notice that 25 can be rewritten as [tex]\(5^2\)[/tex], so:
[tex]\[
\log_5(25) = \log_5(5^2)
\][/tex]
5. Use the Power Rule Again:
Applying the power rule once more:
[tex]\[
\log_5(5^2) = 2 \cdot \log_5(5)
\][/tex]
Since [tex]\(\log_5(5) = 1\)[/tex] (because anything log_b(b) is 1):
[tex]\[
2 \cdot \log_5(5) = 2 \cdot 1 = 2
\][/tex]
6. Substitute Back:
Now substitute back to our previous expression:
[tex]\[
-1 \cdot \log_5(25) = -1 \cdot 2 = -2
\][/tex]
Therefore, the value of [tex]\(\log_5 \frac{1}{25}\)[/tex] is [tex]\(-2\)[/tex].
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