Find the Exact Value log base 25 of 1/5

$\log_{25} (\frac{1}{5})$

Rewrite as an equation.

$\log_{25} (\frac{1}{5}) = x$

Rewrite $\log_{25} (\frac{1}{5}) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$25^{x} = \frac{1}{5}$

Create expressions in the equation that all have equal bases.

${(5^{2})}^{x} = 5^{- 1}$

Rewrite ${(5^{2})}^{x}$ as $5^{2 x}$ .

$5^{2 x} = 5^{- 1}$

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$2 x = - 1$

Solve for $x$ .

$x = - \frac{1}{2}$

The variable $x$ is equal to $- \frac{1}{2}$ .

$- \frac{1}{2}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$

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