Problem 121 - College Algebra Essentials

Question

Determine whether 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

The statement makes sense as it correctly characterizes the properties of exponents and logarithms.

Step 1: Analyze the exponential expression

The expression given is \(b^{m}+b^{n}\). Recall the properties of exponents, you will note that it's only possible to simplify \(b^{m} * b^{n}\) not \(b^{m} + b^{n}\) as \(b^{m} * b^{n} = b^{m+n}\). Therefore, the first part of the statement which says 'cannot simplify the expression \(b^{m} + b^{n}\) by adding exponents' makes sense.

Step 2: Understand the property of logarithm

The logarithm of the product of two numbers is the sum of the logarithms of the numbers. Mathematically, this can be written as: \(\log{ab} = \log{a} + \log{b}\). However, there is no property that allows the logarithm of a sum to be simplified in a similar manner. So, the second part 'there is no property for the logarithm of a sum' also makes sense.

Step 3: Determine the sense of the statement

Both parts of the statement are mathematically accurate in terms of the properties of exponents and logarithms. Thus, the statement as a whole makes sense.

Key Concepts

Properties of Exponents

Properties of exponents are crucial in simplifying expressions involving powers. One of the key properties is that when you multiply like bases with different exponents, you add the exponents. For example, consider the expression \(b^m \cdot b^n\). Here, the rule applies that \(b^m \cdot b^n = b^{m+n}\).

This is because when multiplying powers with the same base, you're essentially adding up the number of times the base is multiplied by itself.

However, the same property does not apply to addition. In the case of \(b^m + b^n\), there's no similar simplification. You can't just add exponents together unless the terms have the same exponent. This is a frequent misunderstanding for many students. Remember, only multiplication of powers allows for simplified computation of exponents in this way.

Logarithm Sum

Logarithms have their own set of properties that often parallel those of exponents, but with important differences.

One key property involves the logarithm of products. If you have a product like \(ab\), the logarithm property states that \(\log(ab) = \log(a) + \log(b)\). This is because logs convert multiplication into addition, reflecting their nature as the inverse of exponentiation.

• This property does not work for sums – for instance, \(\log(a + b)\) cannot be simply split into \(\log(a) + \log(b)\).
• The absence of such a property for sums often needs emphasis, as it prevents common calculation errors.

It's crucial to remember these distinctions between sums and products when working with logarithmic expressions, helping avoid confusion in problem-solving.

Simplifying Expressions

Simplifying expressions involves various strategies depending on the operation type. For expressions with exponents, it's important to apply the correct rules based on multiplication or addition.

Let's break this down:

• When dealing with multiplication, as mentioned earlier, you can simplify \(b^m \cdot b^n\) to \(b^{m+n}\). This method neatly reduces the complexity of the expression under multiplication.
• However, if faced with an addition like \(b^m + b^n\), you can't simplify by merely combining exponents. They remain separate unless factoring is a viable simplification.

By understanding these principles, you can effectively distinguish which rules to use in each scenario, aiding your overall mathematical fluency and reducing errors in calculations.