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Answer :
To solve the given equation [tex]\(\log _3 \frac{1}{25}=\square \log _3 5\)[/tex], we can follow these steps:
1. Rewrite the Fraction as an Exponent:
We begin by rewriting [tex]\(\frac{1}{25}\)[/tex] as an exponent of 5. Notice that:
[tex]\[
\frac{1}{25} = 25^{-1}
\][/tex]
Since [tex]\(25\)[/tex] equals [tex]\(5^2\)[/tex], we have:
[tex]\[
25 = 5^2 \implies \frac{1}{25} = (5^2)^{-1} = 5^{-2}
\][/tex]
2. Express the Logarithm Using Power Rules:
With [tex]\(\frac{1}{25}\)[/tex] rewritten as [tex]\(5^{-2}\)[/tex], the logarithmic expression becomes:
[tex]\[
\log_3\left(\frac{1}{25}\right) = \log_3(5^{-2})
\][/tex]
Apply the power rule of logarithms, which states that [tex]\(\log_b(a^n) = n \cdot \log_b(a)\)[/tex]:
[tex]\[
\log_3(5^{-2}) = -2 \cdot \log_3(5)
\][/tex]
3. Compare Both Sides of the Equation:
We now compare this result to the original equation:
[tex]\[
\log_3 \frac{1}{25}=\square \log_3 5
\][/tex]
Substitute what we found:
[tex]\[
-2 \cdot \log_3(5) = \square \cdot \log_3(5)
\][/tex]
4. Deduce the Value of ⬜:
Since the [tex]\(\log_3(5)\)[/tex] on both sides can be equated, we find:
[tex]\[
\square = -2
\][/tex]
Thus, the value of [tex]\(\square\)[/tex] is [tex]\(-2\)[/tex].
1. Rewrite the Fraction as an Exponent:
We begin by rewriting [tex]\(\frac{1}{25}\)[/tex] as an exponent of 5. Notice that:
[tex]\[
\frac{1}{25} = 25^{-1}
\][/tex]
Since [tex]\(25\)[/tex] equals [tex]\(5^2\)[/tex], we have:
[tex]\[
25 = 5^2 \implies \frac{1}{25} = (5^2)^{-1} = 5^{-2}
\][/tex]
2. Express the Logarithm Using Power Rules:
With [tex]\(\frac{1}{25}\)[/tex] rewritten as [tex]\(5^{-2}\)[/tex], the logarithmic expression becomes:
[tex]\[
\log_3\left(\frac{1}{25}\right) = \log_3(5^{-2})
\][/tex]
Apply the power rule of logarithms, which states that [tex]\(\log_b(a^n) = n \cdot \log_b(a)\)[/tex]:
[tex]\[
\log_3(5^{-2}) = -2 \cdot \log_3(5)
\][/tex]
3. Compare Both Sides of the Equation:
We now compare this result to the original equation:
[tex]\[
\log_3 \frac{1}{25}=\square \log_3 5
\][/tex]
Substitute what we found:
[tex]\[
-2 \cdot \log_3(5) = \square \cdot \log_3(5)
\][/tex]
4. Deduce the Value of ⬜:
Since the [tex]\(\log_3(5)\)[/tex] on both sides can be equated, we find:
[tex]\[
\square = -2
\][/tex]
Thus, the value of [tex]\(\square\)[/tex] is [tex]\(-2\)[/tex].
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