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Chapter 4: Problem 81

Use the properties of logarithms to condense the expression.$$\ln x-2[\ln (x+2)+\ln (x-2)]$$.

Short Answer

Expert verified
The condensed form of the expression is \( \ln{[\frac{x}{(x^2 - 4)^2}]} \).

Step by step solution

01

Apply the Addition Law

First, apply the addition law of logarithms to condense \( \ln(x+2) + \ln(x-2) \) into a single logarithm. This gives \( 2\ln{(x^2 - 4)} \). The equation now becomes \( \ln{x} - 2\ln{(x^2 - 4)} \)
02

Apply the Subtraction Law

Next, apply the subtraction law of logarithms to condense \( \ln{x} - 2\ln{(x^2 - 4)} \) into a single logarithm. This step can be a bit tricky - before applying the subtraction law, first apply the property \( \log{a^n} = n\log{a} \) to rewrite \( 2\ln{(x^2 - 4)} \) as \( \ln{(x^2 - 4)^2} \), which simplifies to \( \ln{(x^2 - 4)^2} \). The expression now becomes \( \ln{x} - \ln{(x^2 - 4)^2} \).
03

Final Simplification

Finally, apply the subtraction law once more to combine \( \ln{x} - \ln{(x^2 - 4)^2} \) into a single logarithm: \( \ln{[\frac{x}{(x^2 - 4)^2}]} \). This is the final condensed form of the given expression.

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Key Concepts

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

Logarithm Condensation
Understanding logarithm condensation is essential for simplifying complex logarithmic expressions into a more manageable form. Essentially, it's about combining several logarithmic terms into a single term by using specific logarithmic laws.

For example, in the exercise provided, we combined \( \ln(x+2) + \ln(x-2) \) by leveraging the addition law of logarithms. This property tells us that the sum of two logarithms with the same base can be condensed into a single logarithm by multiplying their inside numbers (the numbers that the log is applied to). This manipulation is often the first step in simplifying an expression as it makes further steps more straightforward.
Logarithm Laws
There are three main logarithm laws that enable the condensation and expansion of logarithmic expressions: the product law, the quotient law, and the power law.

  • The product law states that \( \log_{a}(xy) = \log_{a}(x) + \log_{a}(y) \)鈥攁llowing us to condense the logs of two products into a single log.
  • The quotient law asserts that \( \log_{a}(\frac{x}{y}) = \log_{a}(x) - \log_{a}(y) \), permitting us to condense a division into a subtraction of logs.
  • The power law indicates that \( \log_{a}(x^n) = n\cdot \log_{a}(x) \), showing how to turn the log of an exponent into a multiplication.

These laws were instrumental in the step-by-step solution, enabling the transformation of the initial complex expression into a single logarithmic term.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is a logarithm to the base of the number \(e\), approximately equal to 2.71828. It's a special type of logarithm because the number \(e\) has many unique properties, especially in the realm of calculus and exponential growth.

In the context of our example, we interact with the natural logarithm when we condense the expression. The properties of the natural logarithm work the same as those of logarithms with any other base, so all the logarithm laws apply here as well. This universality makes transitioning from working with one logarithmic base to another quite seamless.
Logarithmic Properties
Beyond the logarithm laws, there are additional logarithmic properties that are helpful when working with these functions. Key properties include:

  • The logarithm of one is always zero: \(\log_{a}(1) = 0\) no matter what the base \(a\) is (provided it's positive and not equal to 1).
  • Logarithms can only take positive real numbers as their input, reflecting the fact that you can't have the log of a negative number or zero in real numbers.
  • The inverse property links exponentiation and logarithms: \(a^{\log_{a}(x)} = x\).
  • Change of base formula allows the conversion of logarithms to different bases for easier calculation or comparison.

These properties are immensely helpful for analyzing and solving logarithmic equations, as they provide a framework for manipulation and understanding of these mathematical functions.

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Most popular questions from this chapter

An exponential model of the form \(y=a b^{x}\) can be rewritten as a natural exponential model of the form _________.

Use the regression feature of a graphing utility to find a power model \(y=a x^{b}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(1,2.0),(2,3.4),(5,6.7),(6,7.3),(10,12.0)$$

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The table shows the numbers of single beds \(B\) (in thousands) on North American cruise ships from 2007 through 2012. (Source: Cruise Lines International Association) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Beds, } B \\\\\hline 2007 & 260.0 \\\2008 & 270.7 \\\2009 & 284.8 \\\2010 & 307.7 \\\2011 & 321.2 \\\2012 & 333.7 \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find a linear model, an exponential model, and a logarithmic model for the data and identify the coefficient of determination for each model. Let \(t\) represent the year, with \(t=7\) corresponding to 2007 (b) Which model is the best fit for the data? Explain. (c) Use the model you chose in part (b) to predict the number of beds in 2017 . Is the number reasonable?
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