Rewrite each equation as requested. (a) Rewrite as a logarithmic equation. \[ 2^{5}=32 \] (b) Rewrite as an exponential equation. \[ \log _{5} \frac{1}{5}=-1 \] (a) ${ }^{\log } \square \square=\square$ (b) $\square=\square$

Step 1 ：Rewrite each equation as requested.

Step 2 ：For part (a), we are given an exponential equation and we need to rewrite it as a logarithmic equation. The general form of a logarithmic equation is \( \log_b(a) = c \), which is equivalent to the exponential equation \( b^c = a \). So, we need to identify the base (b), the exponent (c), and the result (a) in the given exponential equation and rewrite it in the form of a logarithmic equation.

Step 3 ：The given exponential equation is \( 2^5 = 32 \). Here, the base (b) is 2, the exponent (c) is 5, and the result (a) is 32. So, the equivalent logarithmic equation is \( \log_2(32) = 5 \).

Step 4 ：For part (b), we are given a logarithmic equation and we need to rewrite it as an exponential equation. The general form of an exponential equation is \( b^c = a \), which is equivalent to the logarithmic equation \( \log_b(a) = c \). So, we need to identify the base (b), the result (a), and the logarithm (c) in the given logarithmic equation and rewrite it in the form of an exponential equation.

Step 5 ：The given logarithmic equation is \( \log_5(1/5) = -1 \). Here, the base (b) is 5, the result (a) is 1/5, and the logarithm (c) is -1. So, the equivalent exponential equation is \( 5^{-1} = 1/5 \).

Step 6 ：Final Answer: \n(a) The equivalent logarithmic equation is \( \boxed{\log_{2}{32} = 5} \)\n(b) The equivalent exponential equation is \( \boxed{5^{-1} = \frac{1}{5}} \)