Express the equation in logarithmic form:
(a) $4^3=64$ is equivalent to $\log A=B$.
$$A=$$
and
$$B=$$
(b) ${10}^{-4}=0.0001$ is equivalent to ${\log }_{10}C=D$.
$$C=$$
and
$$D=$$

Step 1 ：Express the equation in logarithmic form: $4^3=64$ is equivalent to $\log A=B$.

Step 2 ：Identify the base, exponent, and result in the equation $4^3=64$. The base is 4, the exponent is 3, and the result is 64.

Step 3 ：Convert the equation into logarithmic form: ${\log }_{4}64=3$.

Step 4 ：Assign the values to A and B: $A=64$ and $B=3$.

Step 5 ：Express the equation in logarithmic form: ${10}^{-4}=0.0001$ is equivalent to ${\log }_{10}C=D$.

Step 6 ：Identify the base, exponent, and result in the equation ${10}^{-4}=0.0001$. The base is 10, the exponent is -4, and the result is 0.0001.

Step 7 ：Convert the equation into logarithmic form: ${\log }_{10}0.0001=-4$.

Step 8 ：Assign the values to C and D: $C=0.0001$ and $D=-4$.

Step 9 ：Final Answer: For the first part, $\boxed{{\displaystyle A=64,B=3}}$. For the second part, $\boxed{{\displaystyle C=0.0001,D=-4}}$.