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 \[ D= \]
Solution
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, \(\boxed{A = 64, B = 3}\). For the second part, \(\boxed{C = 0.0001, D = -4}\).
