Problem 35 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The expression \( \ln \sqrt{x^{2}+1} \) can be written as a multiple of logarithms as \( \frac{1}{2}\ln({x^{2}+1}) \).

Step 1: Recognizing the Root

Notice that our expression starts with a natural logarithm (ln) of a square root. In logarithms, a square root can be represented as a fraction of 1/2. So, the expression can be rewritten as \( \frac{1}{2}\ln ({x^{2}+1}) \).

Step 2: Final Result

Since we're asked to write the expression as a sum, difference, or multiple of logarithms, we can stop here. The expression \( \frac{1}{2}\ln({x^{2}+1}) \) satisfies this condition, as it's half of the natural logarithm of (\(x^{2}+1\)).

Key Concepts

Properties of Logarithms

Logarithms have several properties that make them incredibly useful in simplifying expressions, solving equations, and analyzing mathematical relationships. These properties allow us to break down complex logarithmic expressions into simpler forms, which can be more easily understood and manipulated. Here are some fundamental properties:

• Product Rule: \(\log_b(M \cdot N) = \log_b M + \log_b N\). This rule states that the logarithm of a product is the sum of the logarithms of the factors.
• Quotient Rule: \(\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N\). According to this rule, the logarithm of a quotient is the difference of the logarithms.
• Power Rule: \(\log_b(M^n) = n \cdot \log_b M\). This states that the logarithm of a power can be expressed as the exponent times the logarithm of the base.
• Change of Base Formula: \(\log_b M = \frac{\log_k M}{\log_k b}\), which allows you to change the base of a logarithm.

Understanding these properties is crucial for simplifying logarithmic expressions effectively. In the given problem, the power rule is applied to convert the square root into the exponent \(\frac{1}{2}\), making the expression more straightforward to deal with.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a special type of logarithm where the base is the mathematical constant \(e\), approximately equal to 2.71828. It's called "natural" because it arises naturally in mathematics, particularly in calculus and complex analysis.

• Natural logarithms have properties similar to other logarithms, with \(\ln(e) = 1\) since \(e^1 = e\).
• This logarithm is used extensively in mathematical modeling of real-world phenomena, including exponential growth and decay, as well as compound interest calculations.

One distinguishing feature of natural logarithms is their appearance in problems involving continuous growth or decay. For example, in biology or finance, processes that evolve naturally over time can often be symbolized through expressions involving \(\ln\). This makes them prevalent in various scientific and engineering fields.

Simplifying Expressions

Simplifying expressions is the process of altering them into a more manageable form without changing their value. For logarithmic expressions, this often involves leveraging logarithmic properties to break the expression down.

• In the example exercise, the expression \(\ln \sqrt{x^2 + 1}\) is simplified using the power rule of logarithms, which converts the square root (\(\sqrt{}\)) into an exponent (\(\frac{1}{2}\)).
• The expression then becomes \(\frac{1}{2}\ln(x^2 + 1)\). This form is much simpler because it represents a multiplication of a constant with a natural logarithm, rather than dealing with the complexity of a square root.

By using the properties of logarithms, expressions that initially seem complicated can often be reduced to much simpler forms. This not only makes further mathematical operations easier but also helps in better understanding the underlying relationships within the problem.