Problem 67 - College Algebra Essentials

Question

Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(\frac{1}{3}\left[2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right]\)

Step-by-Step Solution

Answer

The condensed form of the given expression is \( \ln \left[ \left(\frac{(x+5)}{\sqrt[3]{x(x^2-4)}} \right)^2 \right] \)

Step 1: Simplify the Expression inside Brackets

First of all, apply the logarithm rules inside the brackets. So we get: \( \frac{1}{3} \left[\ln (x+5)^2 -\ln x - \ln \left(x^{2}-4\right) \right] \)

Step 2: Apply the Logarithm Rules Further

Use the quotient rule to further condense the expression. Remember the rule states that \ln a - \ln b = \ln (a/b). So our expression becomes: \(\frac{1}{3} \ln \frac{(x+5)^2}{x(x^2-4)}\)

Step 3: Remove the Coefficient of the Logarithm

We have to write the expression as a single log whose coefficient is 1. To remove the coefficient of 1/3, remember that a coefficient on a log can be transferred as an exponent inside the log according to the product rule. So our expression becomes: \(\ln \left[\frac{(x+5)^2}{x(x^2-4)} \right]^{1/3} \).

Step 4: Simplify the Expression

The final step is to simplify the expression inside the third root which gives: \(\ln \left[ \left(\frac{(x+5)}{\sqrt[3]{x(x^2-4)}} \right)^2 \right] \)

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial when it comes to working with logarithmic expressions. Logarithms, the inverse operations of exponentiation, have unique characteristics that allow us to manipulate and simplify complex expressions.

Some key properties include the product rule, which states that the logarithm of a product is the sum of the logarithms (\forall a,b > 0, \(\log(ab) = \log(a) + \log(b)\)), and the quotient rule, where the logarithm of a quotient is the difference of the logarithms (\forall a,b > 0, \(\log(\frac{a}{b}) = \log(a) - \log(b)\)). Another important property is the power rule, which allows us to move exponents outside the logarithm (\forall a > 0, \forall k \in \mathbb{R}, \(\log(a^k) = k\cdot\log(a)\)).

These rules are immensely helpful in transforming and condensing logarithmic expressions and are employed extensively when solving logarithmic equations.

Logarithm Rules

Developing a strong grasp of logarithm rules is essential for anyone looking to master mathematics involving logarithms. These rules are formulas that describe how logarithmic functions behave with different operations, making it easier to work with them.

Key Rules for Simplifying Logarithms

• Product Rule: \forall a,b > 0, \(\log(a\bn + \log(b) = \log(ab)\)). This rule simplifies the multiplication of two log terms into a single log term by adding them.
• Quotient Rule: \forall a,b > 0, \(\log(\frac{a}{b}) = \log(a) - \log(b)\)). This is useful for dividing two log terms.
• Power Rule: \forall a > 0, \forall k \forall k \in \mathbb{R}, \(\log(a^k) = k\cdot\log(a)\)), allowing exponents to be pulled out of the logarithm as coefficients.

By applying these rules appropriately, you can rewrite logarithmic expressions in a form that often makes them easier to evaluate or further manipulate.

Simplifying Logarithmic Expressions

The ability to simplify logarithmic expressions is a fundamental skill in mathematics, helping to make complex log expressions more understandable and easier to work with. The goal when simplifying is to write a log expression in its most compact form, and this often involves using a combination of the logarithm rules mentioned earlier.

To simplify a logarithmic expression, one might combine log terms using the product or quotient rule, convert log terms into exponents with the power rule, and apply other algebraic manipulations to reduce the expression to a simpler form.

For instance, the exercise provided showcases the process of simplifying a compound logarithmic expression into a single logarithm. Each transformation applies different log rules strategically to achieve a simpler form, eventually removing coefficients and combining terms until the expression stands as a single, neat logarithmic entity. This not only streamlines the expression but also sets the stage for possible further evaluation without the use of a calculator.