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Write the logarithmic equation in exponential form.

[tex]
\[
\begin{aligned}
\log _5 \frac{1}{25} & = -2 \\
5^{-2} & = \frac{1}{25}
\end{aligned}
\]
[/tex]

Answer :

We start with the logarithmic equation:

[tex]$$
\log_5 \frac{1}{25} = -2.
$$[/tex]

Recall the definition of logarithms: if

[tex]$$
\log_b (a) = c,
$$[/tex]

then the equivalent exponential form is

[tex]$$
a = b^c.
$$[/tex]

Using this definition with [tex]$a = \frac{1}{25}$[/tex], [tex]$b = 5$[/tex], and [tex]$c = -2$[/tex], we rewrite the equation as:

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

This is the logarithmic equation in exponential form.

To check, we note that:

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

which confirms that the conversion is correct.

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