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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. The product, quotient, and power rules for logarithms directly relate to similar rules for operations with exponents because logarithms are indeed exponents. Therefore, the reasoning is valid and in line with the mathematical principles of logarithms and exponents.

Step by step solution

01

Understanding the statement

The statement is saying that there is a relationship between the rules of operations with logarithms and those with exponents due to the fact that logarithms are exponents. Logarithms indeed are exponents, with a logarithm of a number to a certain base being the exponent that the base must be raised to in order to obtain that number.
02

Comparing the product, quotient, and power rules for logarithms and exponents

The product rule for logarithms states that the logarithm of a product, \( \log_{b}(xy) \), is equal to the sum of the logarithms: \( \log_{b}(x) + \log_{b}(y) \). The quotient rule states that the logarithm of a quotient, \( \log_{b}(x/y) \), is equal to the difference of the logarithms: \( \log_{b}(x) - \log_{b}(y) \). Then, the power rule, \( \log_{b}(x^n) \), is equal to \( n \cdot \log_{b}(x) \), the exponent times the logarithm. The rules for exponents are similar, with the product of the same base resulting in an addition of the exponents, the quotient resulting in a subtraction of the exponents, and the power of an exponent resulting in a multiplication of the exponents.
03

Validate the reasoning

Given the comparison of the product, quotient, and power rules for logarithms and exponents, we can say the reasoning in the statement is sound and the statement does make sense. Because logarithms are exponents, the rules that apply to operations with exponents reflect directly into the properties of logarithms, making the rules for these operations similar.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Exponents
Exponents are a way to express repeated multiplication of the same number by itself. For example, if you have \(2^3\), this means you multiply 2 by itself a total of three times: \(2 \times 2 \times 2 = 8\). Exponents simplify expressions by avoiding long multiplication chains.There are different rules for exponents that make calculations straightforward. Some key properties include:
  • **Zero exponent rule**: Any non-zero number raised to the power of zero is 1, such as \(5^0 = 1\).
  • **Negative exponent rule**: A negative exponent indicates a reciprocal. For example, \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\).
Easy to use and understand, these properties help in breaking down complex exponential expressions into manageable pieces.
Exploring the Product Rule
The product rule is one of the basic tools for working with logarithms and exponents. When dealing with exponents, it states that when you multiply two powers with the same base, you add their exponents.For instance, \(a^m \times a^n = a^{m+n}\). If you had \(2^3 \times 2^4\), it would equal \(2^{3+4} = 2^7 = 128\). When applying the product rule to logarithms, it states, \(\log_b(xy) = \log_b(x) + \log_b(y)\). This shows how the sum of logs corresponds to the product of their components. This rule is a reflection of how exponents add when bases are multiplied, cementing the close relationship between the two concepts.
Unpacking the Quotient Rule
The quotient rule helps decipher expressions involving division in both exponents and logarithms. For exponents, the rule is simple: when dividing powers with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\).As an example, consider \(3^5 / 3^2\). According to the rule, this simplifies to \(3^{5-2} = 3^3 = 27\).The quotient rule for logarithms offers a parallel principle: \(\log_b(x/y) = \log_b(x) - \log_b(y)\). This subtractive relation resonates with the exponents' approach, providing clarity and simplifying calculations. Utilizing this rule efficiently can make solving logarithmic problems both quicker and more intuitive.
Delving into the Power Rule
The power rule is pivotal in pushing forward both logarithmic and exponential expressions. For exponents, if a power is raised to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).Take \((2^3)^2\), which is simplified as \(2^{3 \cdot 2} = 2^6 = 64\). This shows how exponents interact through multiplication when nested.In logarithms, the power rule indicates: \(\log_b(x^n) = n \cdot \log_b(x)\). This transformation marries the exponent and the logarithm, allowing expressions to be tackled with a structured approach, streamlining complex calculations to simpler steps and showcasing the intrinsic connections between logarithms and exponents.

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Most popular questions from this chapter

The \(p H\) scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to \(14 .\) A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a pH greater than 7. The lower the \(p H\) below 7 , the more acidic is the solution. Each whole-number decrease in \(p H\) represents a tenfold increase in acidity. (GRAPH CAN'T COPY). The \(p H\) of a solution is given by $$\mathrm{pH}=-\log x$$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve. Express answers as powers of \(10 .\) a. Normal, unpolluted rain has a pH of about 5.6. What is the hydrogen ion concentration? b. An environmental concern involves the destructive effects of acid rain. The most acidic rainfall ever had a \(\mathrm{pH}\) of \(2.4 .\) What was the hydrogen ion concentration? c. How many times greater is the hydrogen ion concentration of the acidic rainfall in part (b) than the normal rainfall in part (a)?

Use the exponential decay model for carbon- \(14, A=A_{0} e^{-0.000121 t}\) Skeletons were found at a construction site in San Francisco in \(1989 .\) The skeletons contained \(88 \%\) of the expected amount of carbon-14 found in a living person. In \(1989,\) how old were the skeletons?

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln (x-2)-\ln (x+3)=\ln (x-1)-\ln (x+7)$$

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=100(4.6)^{x}$$

The exponential models describe the population of the indicated country, \(A,\) in millions, t years after \(2010 .\) Use these models to solve Exercises \(1-6\). \begin{aligned} &\text { India } \quad A=1173.1 e^{0.008 t}\\\ &\text { Iraq } \quad A=31.5 e^{0.019 t}\\\ &\text { Japan } \quad A=127.3 e^{-0.006 t}\\\ &\text { Russia } \quad A=141.9 e^{-0.005 t} \end{aligned} Which countries have a decreasing population? By what percentage is the population of these countries decreasing each year?
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