Question

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \left(x \sqrt[3]{x^{2}+1}\right) $$

Short answer

\ln (x) + \frac{1}{3} \ln (x^{2}+1)

Step-by-step solution

Apply Product Rule

Use the product rule of logarithms which states that \(\ln(ab) = \ln(a) + \ln(b)\). Here, \(a=x\) and \(b=\sqrt[3]{x^{2}+1}\). So, the given expression becomes \(\ln (x) + \ln (\sqrt[3]{x^{2}+1})\).

Apply Power Rule

Next, use the power rule of logarithms which states \(\ln(a^{n}) = n \cdot \ln(a)\). Here, \(a=x^{2}+1\) and \(n=1/3\) for the second term. So, it simplifies to \(\ln (x) + \frac{1}{3} \ln (x^{2}+1) \).

Key concepts

Logarithm Product Rule

When working with logarithms, combining and simplifying expressions is a fundamental skill. One of the primary tools for this is the logarithm product rule, which helps break down the logs of products into simpler parts. This rule states that the log of a product is equal to the sum of the logs of its factors. Mathematically, this can be represented as \( \text{log}_b(mn) = \text{log}_b(m) + \text{log}_b(n) \).

For example, if we have \( \text{log}_2(3 \times 4) \), using the product rule it becomes \( \text{log}_2(3) + \text{log}_2(4) \). This property proves incredibly useful when dealing with expressions involving products within a logarithm, as it makes them simpler to handle and solve.

Logarithm Power Rule

Similarly essential in the toolbox for simplifying logarithmic expressions is the logarithm power rule. This rule comes into play when you're working with expressions where a logarithm is taken to a power. According to this rule, you can move the exponent of the argument to the front of the log expression as a multiplier. The formula is written as \( \text{log}_b(a^n) = n \times \text{log}_b(a) \).

Imagine dealing with \( \text{log}_3(9^2) \). By the power rule, you can change it to \( 2 \times \text{log}_3(9) \), which simplifies the expression. It's particularly useful for simplifying and solving equations that include terms raised to a power within a logarithm.

Natural Logarithms

Diving into specific types of logarithms, natural logarithms are a special category denoted by \( \text{ln}(x) \) and are based on the natural constant \( e \), approximately equal to 2.718. The natural logarithm of a number is the power to which \( e \) must be raised to obtain that number. For example, if \( e^y = x \), then \( \text{ln}(x) = y \).

Natural logarithms are particularly important in higher mathematics and appear in various applications across physics, engineering, and economics. They are the inverse operation of taking \( e \) to a power, and hence, they play a crucial role in solving exponential growth problems and finding time constants in decay processes.

Simplifying Logarithmic Expressions

When faced with complex logarithmic expressions, the ultimate goal is to simplify them into a form that's more manageable. This process often involves utilizing the product rule, power rule, and other properties of logarithms. The ability to simplify effectively allows for easier computation, comparison, and in many cases, can reveal the underlying patterns or solutions within logarithmic equations.

A simplification task might be transforming a single log with a complex argument into a series of logs added or subtracted from each other. Or it might be taking a compound expression and peeling away the layers by using these rules repeatedly. Simplifying is not only a skill but also an art that, when mastered, yields deep insight into the behavior of logarithmic functions and their characteristics in different mathematical scenarios.