Problem 121 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I cannot simplify the expression \(b^{m}+b^{n}\) by adding exponents, there is no property for the logarithm of a sum.

Step-by-Step Solution

Answer

Yes, the provided statement makes sense. There is no property of exponents that allows for the simplification of \(b^{m} + b^{n}\) by adding exponents. Similarly, there is no logarithm property for the simplification of the logarithm of a sum.

Step 1: Identify Given Statement

The given statement is: 'Because \(b^{m}+b^{n}\) cannot be simplified by adding exponents, there is no property for the logarithm of a sum.' Analyze this as the first step.

Step 2: Understand Exponent Properties

Exponent addition applies when two exponents with the same base are multiplied, not added. In other words, \(b^{m}*b^{n} = b^{m+n}\), not \(b^{m}+b^{n}\). The rule does not apply in the context of addition.

Step 3: Understand Logarithmic Properties

In the case of logarithms, similar to exponents, the log of a product is the sum of the logs: \(log_{b}(m*n) = log_{b}(m) + log_{b}(n)\). However, there is no property that allows for the logarithm of a sum, i.e., there is no formula where \(log_{b}(m+n)\) can be rewritten as the sum, difference or any other combination of \(log_{b}(m)\) and \(log_{b}(n)\).

Step 4: Compare Statement with Properties

Considering the properties of exponents and logarithms, it is clear that the given statement does make sense. The inability to simplify the expression \(b^{m} + b^{n}\) by adding exponents indeed corroborates the lack of a corresponding property for simplifying the logarithm of a sum.