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Chapter 3: Problem 122

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because logarithms are exponents, the product, quotient, and power rules remind me of properties for operations with exponents.

Short Answer

Expert verified
The statement does make sense. Logarithms are the exponents to which a certain base, must be raised to get a number. Therefore, they fundamentally hold the same properties of exponents, such as the product, quotient, and power rules.

Step by step solution

01

Introduction

Firstly, let us remember the nature and properties of logarithms and exponents. Logarithms and exponents are indeed closely related since a logarithm is basically the exponent to which a certain base must be raised to get a number.
02

Product Rule

Let's look at the Product Rule. For logarithms, the formula for the product rule is \( \log_b(x \cdot y) = \log_b(x) + \log_b(y) \). In relation to exponents, when we multiply like bases, we add the exponents, that is \( b^n \cdot b^m = b^{n+m} \). This shows a parallel between logarithms and exponents.
03

Quotient Rule

Next, let's review the Quotient Rule. For logarithms, the quotient rule is \( \log_b \left( \frac{x}{y} \right) = \log_b(x) - \log_b(y) \). This is similar to the rule for exponents which is \( b^n 梅 b^m = b^{n-m} \). Again, there's a similarity between properties of logarithms and exponents.
04

Power Rule

Finally, the Power Rule. For logarithms, the power rule is \( \log_b(x^n) = n \cdot \log_b(x) \). This has its parallel in the rules for exponents, where \( (b^n)^m = b^{n \cdot m} \). Again, the properties of logarithms resemble the rules of exponents.
05

Conclusion

In all three cases we see that the rules for logarithms and exponents are similar. This shows us that the statement does make sense, since both logarithms and exponents follow similar rules for their operations.

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