Problem 145 - College Algebra

Question

If \(b^{A}=M N, b^{C}=M,\) and \(b^{D}=N,\) what is the relationship among \(A, C,\) and \(D ?\)

Step-by-Step Solution

Answer

The relationship among \(A, C,\) and \(D\) is \(A = C + D\)

Step 1: Understand the properties of exponential functions

In exponential functions, for any base \(b\), \(b^{m}*b^{n} = b^{m+n}\). This property will be helpful while establishing the relationship between \(A, C,\) and \(D\).

Step 2: Writing \(b^{A}\) in terms of \(b^{C}\) and \(b^{D}\)

According to the given equations, \(b^{A} = MN\), \(b^{C} = M\), and \(b^{D} = N\). Therefore, \(b^{A} = b^{C}*b^{D}\). By making use of the properties of exponential functions, this can be rewritten as \(b^{A} = b^{C+D}\). Hence, the required relationship between \(A, C,\) and \(D\) is \(A = C + D\).