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Answer :
To solve the equation [tex]\(5^x = \frac{1}{25}\)[/tex], we need to express both sides as powers of the same base, whenever possible.
1. Rewrite the fraction:
The fraction [tex]\(\frac{1}{25}\)[/tex] can be rewritten as a power of 5. Since [tex]\(25\)[/tex] is equal to [tex]\(5^2\)[/tex], the reciprocal [tex]\(\frac{1}{25}\)[/tex] can be expressed as a power:
[tex]\[
\frac{1}{25} = (5^2)^{-1} = 5^{-2}
\][/tex]
2. Set up the equation with the same base:
We now have the equation:
[tex]\[
5^x = 5^{-2}
\][/tex]
3. Compare the exponents:
Since the bases on both sides of the equation are the same, we can set the exponents equal to each other:
[tex]\[
x = -2
\][/tex]
Thus, the solution to the equation [tex]\(5^x = \frac{1}{25}\)[/tex] is [tex]\(x = -2\)[/tex].
1. Rewrite the fraction:
The fraction [tex]\(\frac{1}{25}\)[/tex] can be rewritten as a power of 5. Since [tex]\(25\)[/tex] is equal to [tex]\(5^2\)[/tex], the reciprocal [tex]\(\frac{1}{25}\)[/tex] can be expressed as a power:
[tex]\[
\frac{1}{25} = (5^2)^{-1} = 5^{-2}
\][/tex]
2. Set up the equation with the same base:
We now have the equation:
[tex]\[
5^x = 5^{-2}
\][/tex]
3. Compare the exponents:
Since the bases on both sides of the equation are the same, we can set the exponents equal to each other:
[tex]\[
x = -2
\][/tex]
Thus, the solution to the equation [tex]\(5^x = \frac{1}{25}\)[/tex] is [tex]\(x = -2\)[/tex].
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