Problem 39 - College Algebra Essentials

Question

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log \left[\frac{10 x^{2} \sqrt[3]{1-x}}{7(x+1)^{2}}\right]\)

Step-by-Step Solution

Answer

The expanded and simplified form of the given logarithmic expression is \(1 + 2\log(x) + \frac{1}{3}\log(1-x) - \log_{10}(7) - 2\log(x+1)\)

Step 1: Using the Quotient Rule

The quotient rule states that the logarithm of a quotient is the difference of the logarithms. This rule can be used to break down the expression. So, \(\log \left[\frac{10 x^{2} \sqrt[3]{1-x}}{7(x+1)^{2}}\right]\) becomes \(\log (10x^{2} \sqrt[3]{1-x}) - \log (7(x+1)^{2})\)

Step 2: Using the Product Rule

The product rule states that the logarithm of a product is the sum of the logarithms. Thus, the expression \(\log (10x^{2} \sqrt[3]{1-x})\) can be further broken down to \(\log(10) + \log(x^{2}) + \log(\sqrt[3]{1-x})\) and \(\log (7(x+1)^{2})\) can be broken down to \(\log(7) + \log((x+1)^{2})\). Hence, the expression becomes \(\log(10) + \log(x^{2}) + \log(\sqrt[3]{1-x}) - (\log(7) + \log((x+1)^{2}))\)

Step 3: Using the Square Root and Power Rule

The power rule states that the logarithm of a power is the exponent times the logarithm of the base. It can be used to break down \(\log(x^{2})\) and \(\log((x+1)^{2})\). Hence, the expression becomes \(\log(10) + 2\log(x) + \frac{1}{3}\log(1-x) - (\log(7) + 2\log(x+1))\)

Step 4: Evaluating Logarithmic Expressions

Now that the expression is fully expanded, the values of known logarithms can be filled in. Since \(\log_{10}(10)=1\), and \(\log_{10}(7)\) cannot be simplified further without a calculator, the test expression simplifies to \(1 + 2\log(x) + \frac{1}{3}\log(1-x) - (\log_{10}(7) + 2\log(x+1))\)

Key Concepts

Properties of Logarithms

Logarithms, often seen as the inverse operation of exponentiation, have unique properties that make complex arithmetic problems more manageable. Understanding the properties of logarithms is essential for expanding, condensing, and solving logarithmic expressions.

The key properties include the product rule, the quotient rule, and the power rule. The product rule allows you to turn the logarithm of a product into a sum of logarithms. For example, \(\log(ab) = \log(a) + \log(b)\). The quotient rule helps you transform the logarithm of a fraction into a difference, demonstrated by \(\log(\frac{a}{b}) = \log(a) - \log(b)\). Lastly, the power rule permits taking an exponent out in front of the log, shown as \(\log(a^b) = b\log(a)\).

When you expand logarithmic expressions, you're using these properties to break down the log into simpler parts that can reveal more about the relationship between the numbers and variables involved. Not only does this process aid in solving equations, it also makes it possible to evaluate complex logs without a calculator, provided you encounter familiar base numbers like 10.

Quotient Rule in Logarithms

The quotient rule for logarithms is immensely helpful when working with ratios or fractions within a logarithmic argument. It says that the log of a division between two expressions is the same as the difference of their logs. Mathematically, it's presented as \(\log(\frac{a}{b}) = \log(a) - \log(b)\).

For example, if you have the expression \(\log(\frac{x}{y})\), you can apply the quotient rule to rewrite it as \(\log(x) - \log(y)\), effectively separating the numerator and the denominator into distinct logarithms. This particular property allows for the simplification of complex terms, thus making it easier to expand, condense, or solve logarithmic problems.

Applying the quotient rule not only simplifies expressions but also often sets up the equation for further manipulation using other logarithmic properties, such as the product or power rules.

Product Rule in Logarithms

Another powerful tool in the logarithmic toolbox is the product rule. This rule is handy when you're dealing with multiplication within a log. It asserts that the logarithm of a product is the sum of the logs of the factors: \(\log(ab) = \log(a) + \log(b)\).

This rule is particularly useful for breaking down complex expressions into more digestible parts. By transforming a multiplication into an addition, you also pave the way for easier arithmetic operations and future simplifications. For instance, if your original log term is \(\log(xy)\), you use the product rule to reframe it as \(\log(x) + \log(y)\).

Employing the product rule can lead you to a point where terms might cancel each other out or reveal more straightforward paths to solving for variables. It is a fundamental concept that, combined with other properties of logarithms, ensures a robust approach to deciphering logarithmic expressions.