We are given the equation:

${}^{5}\log\frac{1}{5}$

This represents a logarithm with base $5$, and we're solving for:

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

This can be rewritten as:

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

We know that $\frac{1}{5}$ can be expressed as $5^{-1}$. So we have:

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

Since the bases are the same, we can equate the exponents:

$x = -1$

Therefore:

${}^{5}\log\frac{1}{5} = -1$

Would you like more details or have any questions?

Here are 5 related questions for you:

1. What is the general formula for logarithms in different bases?
2. How can you solve logarithmic equations using exponentiation?
3. What is the change of base formula for logarithms?
4. How do you express fractions as negative exponents?
5. What are some common properties of logarithms?

Tip: When the argument of a logarithm is a reciprocal, its value is often a negative exponent.