We need to solve for the unknowns $b$ and $x$ in the following logarithmic equations: $log_b(5) = \frac{1}{2}$ $log_5(\frac{1}{5}) = x$

1. Problem Description

We need to solve for the unknowns $b$ and $x$ in the following logarithmic equations:

$log_b(5) = \frac{1}{2}$

$log_5(\frac{1}{5}) = x$

2. Solution Steps

First, let's solve for $b$ in the equation $log_b(5) = \frac{1}{2}$. We can rewrite this equation in exponential form:

$b^{\frac{1}{2}} = 5$

To solve for $b$, we can square both sides of the equation:

$(b^{\frac{1}{2}})^2 = 5^2$

$b = 25$

Next, let's solve for $x$ in the equation $log_5(\frac{1}{5}) = x$. We can rewrite $\frac{1}{5}$ as $5^{-1}$:

$log_5(5^{-1}) = x$

Using the property of logarithms that $log_a(a^k) = k$, we have:

$x = -1$

3. Final Answer

$b = 25$

$x = -1$