Problem 79 - Precalculus with Limits

Question

Condensing a Logarithmic Expression In Exercises \(67-82\) , condense the expression to the logarithm of a single quantity. $$\frac{1}{3}\left[2 \ln (x+3)+\ln x-\ln \left(x^{2}-1\right)\right]$$

Step-by-Step Solution

Answer

The condensed logarithmic expression is \(\ln \left(\frac{(x+3)^{\frac{2}{3}} * x^{\frac{1}{3}}}{(x^{2}-1)^{\frac{1}{3}}}\right)\)

Step 1: Distribute the scalar

Distribute the \(\frac{1}{3}\) coefficient to each logarithm in the expression. This will give \(\frac{2}{3} \ln (x+3) + \frac{1}{3} \ln x - \frac{1}{3}\ln (x^{2}-1)\)

Step 2: Use the properties of logarithms

It can be applied that the coefficient of a logarithm is equivalent to an exponent inside the logarithm. Also, the difference of two logarithms is equivalent to the logarithm of a quotient. So the expression can be rewritten as \(\ln ((x+3)^{\frac{2}{3}}) + \ln (x^{\frac{1}{3}}) - \ln ((x^{2}-1)^{\frac{1}{3}})\)

Step 3: Simplify the expression

The sum of two logarithms can be combined into one logarithm via multiplication within the argument, and the difference can be written as division. This gives \(\ln \left(\frac{(x+3)^{\frac{2}{3}} * x^{\frac{1}{3}}}{(x^{2}-1)^{\frac{1}{3}}}\right)\)

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential when it comes to condensing or expanding logarithmic expressions. These properties are the qualities that make logarithms such versatile tools in algebra. One key property is that multiplying constants can be converted into exponents, which can be seen when we take a term like \( a \cdot \log_b(x) \) and rewrite it as \( \log_b(x^a) \). This property is crucial when trying to condense several logarithmic terms into a single logarithm, as it allows the coefficients to transform and become part of the logarithmic argument.

Another important property states that the sum of two logarithms with the same base can be combined into a single logarithm by converting the sum into a multiplication inside the argument, as in \( \log_b(x) + \log_b(y) = \log_b(xy) \). Conversely, the difference of logarithms becomes a division, represented by \( \log_b(x) - \log_b(y) = \log_b(\frac{x}{y}) \). These properties make it possible to simplify complex expressions by condensing several terms into one, thus making the computation much easier and more straightforward.

Logarithmic Functions

Logarithmic functions are the inverse functions of exponential functions and they are used to solve equations where the unknown is the exponent. The notation \( \log_b(x) \) represents the logarithm of \( x \) to the base \( b \), where \( b \) is a positive real number and not equal to 1. The function is defined only for positive \( x \), due to the nature of exponential growth or decay.

A logarithmic function answers the question: 'To what exponent must the base \( b \) be raised to yield \( x \)'? This interpretation is essential when analyzing the behavior of logarithmic functions and allows for a deeper understanding of their application. Logarithmic functions have numerous applications in real-world contexts, including complex calculations involving decibels in sound intensity, the pH scale in chemistry, and Richter scale measurements in seismology.

Simplifying Logarithms

Simplifying logarithms is a process that often involves the use of the properties of logarithms. The goal is to rewrite the logarithmic expression in its simplest form, often as a single logarithm. Doing so not only makes the expressions appear cleaner but also paves the way for easier calculation or further manipulation.

In an exercise such as condensing \( \frac{1}{3}[2 \ln (x+3) + \ln x - \ln (x^2-1)] \), simplification involves distributing the constant, using properties to move exponents, and finally combining terms through multiplication or division. The resulting expression, \( \ln \left(\frac{(x+3)^{\frac{2}{3}} \cdot x^{\frac{1}{3}}}{(x^2-1)^{\frac{1}{3}}}\right) \), is much more manageable. This practice is not only a test of understanding logarithmic properties but it is also excellent exercise in algebraic manipulation, enhancing one's ability to handle complex mathematical expressions.