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Given the logarithmic equation [tex]\log _5 \frac{1}{25} = -2[/tex], convert it to its exponential form. What should be the exponent/power?

A. 5
B. -2
C. [tex]\frac{1}{25}[/tex]
D. 2

Answer :

We start with the definition of the logarithm:

[tex]$$
\log_5 \frac{1}{25} = x \quad \Longleftrightarrow \quad 5^x = \frac{1}{25}.
$$[/tex]

Notice that we can express [tex]$\frac{1}{25}$[/tex] as a power of [tex]$5$[/tex]. Since

[tex]$$
25 = 5^2,
$$[/tex]

it follows that

[tex]$$
\frac{1}{25} = \frac{1}{5^2} = 5^{-2}.
$$[/tex]

Thus, we have

[tex]$$
5^x = 5^{-2}.
$$[/tex]

Since the bases are the same, the exponents must be equal. Therefore,

[tex]$$
x = -2.
$$[/tex]

So, the exponent or power is [tex]$\boxed{-2}$[/tex].

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