Question

In Exercises, write the expression as the logarithm of a single quantity. $$ \frac{1}{3}\left[2 \ln (x+3)+\ln x-\ln \left(x^{2}-1\right)\right] $$

Short answer

The simplified expression as a single logarithmic function is: \( \ln \frac{x^{1}(x+3)^{2}}{x^{2} - 1} \)

Step-by-step solution

Distribute the constant 1/3

Distribute the constant fraction \( \frac{1}{3} \) across the terms within the bracket: \( \frac{2}{3} \ln (x+3) + \frac{1}{3} \ln x - \frac{1}{3} \ln (x^{2} - 1) \)

Apply exponent rule

In logarithms, multiplying the log by a constant is equal to taking the log of the argument raised to that constant. Apply this exponent rule to the modified expression: \( \ln (x+3)^{2/3} + \ln x^{1/3} - \ln (x^{2} - 1)^{1/3} \)

Convert to a single logarithm

Using the properties of logarithms, we can add or subtract the logarithms by multiplying or dividing their quantities respectively. Do this to convert the expression into a single logarithm: \( \ln \frac{(x+3)^{2/3} x^{1/3}}{(x^{2} - 1)^{1/3}} \)

Simplify the expression

Simplify the expression inside the logarithm to reach the final answer: \( \ln \frac{x^{1}(x+3)^{2}}{x^{2} - 1} \)

Key concepts

Exponent Rules

Understanding exponent rules is crucial when working with logarithmic expressions because it allows us to simplify expressions and solve equations with ease. In the context of logarithms, the exponent rule states that multiplying a logarithm by a constant is equivalent to taking the logarithm of the argument raised to that constant. For example, if we have a term like \( a \ln b \), it can be rewritten as \( \ln b^a \). This rule helps simplify expressions by reducing the complexity of calculations.

In the given problem, we see this rule applied when \( \frac{2}{3} \ln (x+3) \) becomes \( \ln (x+3)^{2/3} \), and similarly for the other terms in the expression. This step makes it easier to combine the terms into a single logarithm later on.

Properties of Logarithms

The properties of logarithms are fundamental tools that allow us to manipulate and simplify expressions. These properties include the product, quotient, and power rules of logarithms:

• Product Rule: \( \ln a + \ln b = \ln (ab) \)
• Quotient Rule: \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \)
• Power Rule: \( a \ln b = \ln b^a \)

In this exercise, after applying the exponent rule, we apply these properties to combine the individual logarithms into a single logarithm.

Notice how the quotient rule is applied to the expression \( \ln (x+3)^{2/3} + \ln x^{1/3} - \ln (x^{2} - 1)^{1/3} \) to get a single logarithm: \( \ln \frac{(x+3)^{2/3} x^{1/3}}{(x^{2} - 1)^{1/3}} \). This skill in handling properties of logarithms allows consolidation into simpler expressions, making them easier to evaluate and interpret.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using a combination of exponent rules and the properties of logarithms to reduce expressions to a more manageable form. This not only aids in solving equations involving logarithms but also in the understanding of their behavior.

In our problem, after combining the terms into a single logarithm, further simplification is possible. The expression inside the logarithm, \( \frac{(x+3)^{2/3} x^{1/3}}{(x^{2} - 1)^{1/3}} \), is simplified further by handling powers and potential factoring. The final expression \( \ln \frac{x^{1}(x+3)^{2}}{x^{2} - 1} \) is achieved by canceling out the fractions and simplifying powers. This is a crucial step in solving logarithmic equations efficiently, ensuring that the expression is in its simplest and most interpretable form for further mathematical operations.