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Chapter 3: Problem 105

Describe the product rule for logarithms and give an example.

Short Answer

Expert verified
The product rule for logarithms states that the log of a product is the sum of the logs of its factors, expressed as \( \log_b(xy) = \log_b(x) + \log_b(y) \). For example, \( \log_2(8*4) \) simplifies to \( \log_2(8) + \log_2(4) \), or \(5\).

Step by step solution

01

Description of The Product Rule

The product rule for logarithms states that the logarithm of a product is the sum of the logarithms of its factors. In mathematical terms, this can be written as \( \log_b(xy) = \log_b(x) + \log_b(y) \). Here, \(b\) is the base of the logarithm, and \(x\) and \(y\) are the factors of the product.
02

Example of The Product Rule

Consider an example where \(b=2\), \(x=8\), and \(y=4\). According to the product rule, \( \log_2(8*4) = \log_2(8) + \log_2(4) \). Simplifying the right side gives us \(3 + 2\), which equals \(5\). The left side can also be calculated as \( \log_2(32) \), which also equals \(5\). Hence, the example verifies the product rule of logarithms.

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Key Concepts

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

Logarithmic Identities
Logarithmic identities are fundamental tools in mathematics that make solving equations involving logarithms simpler and more efficient. These identities let us transform logarithmic expressions into equivalent forms that are easier to work with. The product rule is one essential type of logarithmic identity, among others like the quotient rule and power rule.

Understanding these identities is crucial because they help us:
  • Simplify complex logarithmic expressions.
  • Solve logarithmic equations accurately and efficiently.
  • Convert multiplication into addition, which is often easier to manage in algebraic contexts.
Ensuring a clear comprehension of these identities lays the groundwork for more advanced topics in mathematics, such as calculus. Start by practicing different problems, and soon these identities will become second nature.
Logarithms
At its core, a logarithm is the inverse operation of exponentiation. This means that if you have an equation like \( b^x = y \), the logarithm helps you find \( x \) when you know \( b \) (the base) and \( y \) (the result). In symbolic form, this is written as \( x = \log_b(y) \).

Logarithms are powerful because they allow you to:
  • Solve equations where the unknown variable is in the exponent.
  • Transform multiplicative relationships into additive ones (e.g., using the product rule).
  • Measure exponential growth or decay processes, common in sciences and finance.
Logarithms can be found with different bases, but the most common are base 10, known as the common logarithm, and base \( e \), known as the natural logarithm. Developing a solid grasp of logarithms involves understanding their properties and practice in applying their rules in various mathematical contexts.
Logarithm Properties
Logarithm properties are rules that govern how logarithms behave and interact with each other in mathematical operations, providing shortcuts to simplify calculations. Key logarithm properties you should know include:
  • Product rule: as you've seen, \( \log_b(xy) = \log_b(x) + \log_b(y) \).
  • Quotient rule: turns division into subtraction, expressed as \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \).
  • Power rule: allows you to move an exponent in front as a multiplier, \( \log_b(x^k) = k \cdot \log_b(x) \).
Knowing and applying these properties simplifies mathematical problems significantly, enabling quicker computation and deeper insight into the nature of logarithmic expressions. Whether dealing with complex algebraic equations or real-world applications, these rules provide the framework needed to navigate challenges involving logarithms.

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Most popular questions from this chapter

The formula \(A=37.3 e^{0.0095 t}\) models the population of California, \(A,\) in millions, \(t\) years after 2010 . a. What was the population of California in \(2010 ?\) b. When will the population of California reach 40 million?

The formula \(A=25.1 e^{0.0187 t}\) models the population of Texas, \(A\), in millions, \(t\) years after 2010 . a. What was the population of Texas in \(2010 ?\) b. When will the population of Texas reach 28 million?

By 2019 , nearly 1 dollar out of every 5 dollars spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product (GDP) going toward health care from 2007 through 2014 , with a projection for 2019.(GRAPH CAN'T COPY). The data can be modeled by the function \(f(x)=1.2 \ln x+15.7\) where \(f(x)\) is the percentage of the U.S. gross domestic product going toward health care \(x\) years after \(2006 .\) Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \(2008 .\) Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \(18.6 \%\) of the U.S. gross domestic product go toward health care? Round to the nearest year.

Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$e^{4 x-5}-7=11,243$$

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$\ln x+\ln 7$$
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