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Chapter 10: Problem 29

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$ \log _{10}(x+3)+\log _{10}(x-3) $$

Short Answer

Expert verified
\[\begin{equation}\log_{10}(x^2-9)\end{equation}\]

Step by step solution

01

Understanding the Problem

The goal is to combine two logarithmic expressions into a single logarithm. The given expressions are \ \( \log_{10}(x+3) \ \) and \ \( \log_{10}(x-3) \ \).
02

Applying the Product Rule of Logarithms

Use the product rule of logarithms which states that \ \ \[ \log_{b}(a) + \log_{b}(c) = \log_{b}(ac) \ \ \]. Applying this rule, combine \ \( \log_{10}(x+3) \ \) and \ \( \log_{10}(x-3) \ \).
03

Combining the Logarithms

Using the product rule, the expression \ \( \log_{10}(x+3) + \log_{10}(x-3) \ \) becomes \ \( \log_{10}((x+3)(x-3)) \ \).
04

Final Expression

Simplify the expression inside the logarithm if possible. In this case, it simplifies to \ \( x^2 - 9 \ \). Therefore, the combined expression is \ \( \log_{10}(x^2-9) \ \).

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Key Concepts

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

Logarithmic Expressions
Logarithms are powerful tools in mathematics that help us work with exponential relationships by transforming those relationships into additive and subtractive ones. A logarithmic expression involves a logarithm, which is essentially the inverse function of an exponentiation. For example, if we have the expression \(\text{log}_b(a)\), it means 'to what power must \(b\) be raised to produce \(a\)?'. Here, \(b\) is called the base, and \(a\) is the argument. Logarithmic expressions are commonly manipulated using various rules and properties to simplify or combine them, which can make solving equations more straightforward.
Product Rule of Logarithms
One of the most useful properties of logarithms is the product rule. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the factors. Formally, this rule is expressed as \[\text{log}_b(a) + \text{log}_b(c) = \text{log}_b(ac)\]. This rule allows us to combine two logarithmic expressions into a single logarithm, simplifying calculations and making the expressions easier to work with. For instance, if we have the expressions \(\text{log}_{10}(x+3)\) and \(\text{log}_{10}(x-3)\), applying the product rule combines them into \(\text{log}_{10}((x+3)(x-3))\).
Combining Logarithms
Combining multiple logarithmic expressions into a single one can greatly simplify a problem. This process often involves using logarithmic properties like the product rule, as seen in the previous example. After applying the product rule, we often simplify the argument inside the logarithm. For instance, with \(\text{log}_{10}((x+3)(x-3))\), we simplify the expression inside the logarithm: \((x+3)(x-3) = x^2 - 9\). Therefore, \(\text{log}_{10}(x+3) + \text{log}_{10}(x-3)\) simplifies to \(\text{log}_{10}(x^2 - 9)\). By combining logarithms in this way, we turn complex equations into ones that are easier to handle.

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Most popular questions from this chapter

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$ \log _{10} 2=0.3010 \quad \text { and } \quad \log _{10} 9=0.9542 $$ Evaluate each logarithm by applying the appropriate rule or rules from this section. $$ \log _{10} 9^{5} $$

The domain of \(f(x)=a^{x}\) is \((-\infty, \infty),\) while the range is \((0, \infty) .\) Therefore, since \(g(x)=\log _{a} x\) defines the inverse of \(f,\) the domain of \(g\) is _____, while the range of \(g\) is _____.

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$ \log _{10} 2=0.3010 \quad \text { and } \quad \log _{10} 9=0.9542 $$ Evaluate each logarithm by applying the appropriate rule or rules from this section. $$ \log _{10} \frac{2}{9} $$

Find the hydronium ion concentration of the substance with the given pH. Human gastric contents, 2.0

The concentration of a drug injected into the bloodstream decreases with time. The intervals of time \(T\) when the drug should be administered are given by $$T=\frac{1}{k} \ln \frac{C_{2}}{C_{1}}$$ where \(k\) is a constant determined by the drug in use, \(C_{2}\) is the concentration at which the drug is harmful, and \(C_{1}\) is the concentration below which the drug is ineffective. (Source: Horelick, Brindell and Sinan Koont, "Applications of Calculus to Medicine: Prescribing Safe and Effective Dosage," UMAP Module 202.) Thus, if \(T=4,\) the drug should be administered every 4 hr. For a certain drug, \(k=\frac{1}{3}, C_{2}=5,\) and \(C_{1}=2 .\) How often should the drug be administered?
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