Question

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \frac{2 x}{\sqrt{x^{2}-1}} $$

Short answer

The final expression, after using the properties of logarithms to write it as a sum, difference, or multiple of logarithms, is \(\ln(2) + \ln(x) - \frac{1}{2}\ln(x^{2}-1)\)

Step-by-step solution

Break down the logarithm of the division

The logarithm of a quotient is equal to the difference of the logarithms. Therefore, the expression can be written as: \[\ln(2x) - \ln(\sqrt{x^{2}-1})\]

Break down the logarithm of the product

The logarithm of a product is the sum of the logarithms. We could apply these changes to the first term. The expression now becomes: \[\ln(2) + \ln(x) - \ln(\sqrt{x^{2}-1})\]

Handle the square root in the logarithm

We know that the square root is the same as raising to the power of \(\frac{1}{2})\). The power of a logarithm can be extracted as a multiplicative coefficient. The final expression becomes: \[\ln(2) + \ln(x) - \frac{1}{2}\ln(x^{2}-1)\]

Key concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and they play a vital role in various mathematical and real-world applications. At the heart of their behavior is the log function, typically represented as \( \log_b(x) \) or \( \ln(x) \) for the natural logarithm.

For the expression \( \ln \frac{2x}{\sqrt{x^{2}-1}} \), understanding the nature of logarithmic functions helps us to manipulate the equation. The log function only takes positive real numbers as its input and its output can span from negative infinity to positive infinity. Such characteristics become particularly handy when solving equations or rewriting expressions in a more manageable form, especially when we're looking to isolate variables or simplify complex formulas.

When working with logarithmic functions, we can exploit several rules that allow us to break apart or combine logs, making them less intimidating and more computationally feasible. By transforming multiplicative relationships into additive ones, as seen in our exercise, we apply these invaluable logarithmic properties.

Logarithm Rules

Logarithm rules, or laws of logarithms, are essential tools for rewriting and solving logarithmic expressions. Knowing these rules enables us to simplify complex logarithmic expressions into forms that are easier to interpret and calculate.

Here are a few fundamental logarithm rules illustrated in the given exercise:

• Product Rule: The logarithm of a product is equal to the sum of the logarithms of each factor. Mathematically, \( \log_b(xy) = \log_b(x) + \log_b(y) \).
• Quotient Rule: The logarithm of a quotient is the difference of the logarithms. In formula terms, \( \log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) \).
• Power Rule: A logarithm raised to a power can be expressed as the power multiplied by the logarithm. Thus, \( \log_b(x^p) = p \log_b(x) \).

We use these rules step by step to rework the given expression \( \ln \frac{2x}{\sqrt{x^{2}-1}} \) into \( \ln(2) + \ln(x) - \frac{1}{2}\ln(x^{2}-1) \) by first applying the quotient rule, then the product rule, and finally the power rule to handle the square root.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special type of logarithm that uses the number \(e\) (approximately 2.71828) as its base. This constant \(e\) is fundamental in mathematics, particularly in calculus, and is the base of the natural logarithm because of its unique properties related to growth and decay rates.

The natural logarithm shares the same properties as logarithms with other bases. However, because it is so commonly used, it often goes without the base notation and is simply written as \( \ln(x) \). For instance, in the exercise \( \ln \frac{2x}{\sqrt{x^{2}-1}} \), we are implicitly dealing with a logarithm to the base \(e\), enabling us to apply all the logarithm rules we've discussed.

In calculus and other higher-level mathematics, the natural logarithm is especially prized for its derivative and integral properties, which are smoothly behaved and highly predictable. Understanding the natural logarithm and its relationship to \(e\) is crucial for any student wishing to dive deeper into the fascinating world of exponential and logarithmic functions.