Problem 23 - Calculus of a Single Variable

Question

In Exercises 19–28, use the properties of logarithms to expand the logarithmic expression. $$ \ln \left(x \sqrt{x^{2}+5}\right) $$

Step-by-Step Solution

Answer

\(\ln (x) + \frac{1}{2} \ln (x^{2}+5)\

Step 1: Application of Product Rule

The product rule of logarithm states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Hence, \(\ln (ab) = \ln a + \ln b\). Applying the product rule, the given expression \[\ln (x \sqrt{x^{2}+5})\] could be broken down into \[\ln (x) + \ln (\sqrt{x^{2}+5})\]

Step 2: Simplification Using Power Rule

The power rule states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number (i.e. \(\ln(a^n) = n \ln a\)). Looking at the second term from the previous step, the square root can be written as a power of 1/2: \[\ln (\sqrt{x^{2}+5}) = \ln ((x^{2}+5)^{1/2})\] Following the power rule, this could be rewritten as \[\frac{1}{2} \ln (x^{2}+5)\]

Step 3: Final Solution

By combining the results from the two previous steps, the final expanded version of the logarithm expression is: \[\ln (x) + \frac{1}{2} \ln (x^{2}+5)\]

Key Concepts

Product Rule

The product rule is a fundamental property of logarithms. It allows us to simplify the logarithm of a product by breaking it down into the sum of two separate logarithms. This rule is particularly useful in situations where you have a complex expression. Consider any two positive numbers, say \(a\) and \(b\). According to the product rule:

• \(\ln(ab) = \ln(a) + \ln(b)\)

Using this rule can make it easier to work with or integrate logarithmic functions, especially when you have multiple factors involved. For example, if you have an expression such as \(\ln(x \sqrt{x^2+5})\), applying the product rule lets you expand it into \(\ln(x) + \ln(\sqrt{x^2+5})\), making the math more manageable.

Power Rule

The power rule simplifies logarithmic expressions where exponents are present. It states that if you have a logarithm of an exponentiated number, you can bring the exponent down as a multiplier in front of the logarithm. Mathematically, this rule is expressed as:

• \(\ln(a^n) = n \ln(a)\)

This rule is quite handy when dealing with roots or powers. For instance, when you encounter a square root, it’s equivalent to a power of \(1/2\). So, if given \(\ln(\sqrt{x^2 + 5})\), you can rewrite the square root as a power: \(\ln((x^2 + 5)^{1/2})\). You can then apply the power rule to simplify this further into \(\frac{1}{2} \ln(x^2 + 5)\). This step is crucial to reduce the complexity of logarithmic expressions effectively.

Expanding Logarithmic Expressions

Expanding logarithmic expressions involves expressing a single log argument as multiple logs using logarithmic rules. This often makes complicated expressions simpler to analyze or solve. There are key steps to consider:

• Start by identifying if you can use the product rule. Break down the expression into simpler products.
• Analyze each separated term to see if the power rule can ease the expression further.

The example \(\ln(x \sqrt{x^2+5})\) expands first using the product rule into \(\ln(x) + \ln(\sqrt{x^2+5})\). After which, you apply the power rule to the second term to get \(\frac{1}{2} \ln(x^2+5)\). Such expansions are often required in calculus to facilitate integration or derivation.

Properties of Logarithms

Logarithms have several properties that simplify the process of manipulating and restructuring expressions. The main properties include the product, power, and quotient rules, each serving a unique function to aid in the simplification of logarithmic expressions:

• Product Rule: \(\ln(a \cdot b) = \ln(a) + \ln(b)\)
• Power Rule: \(\ln(a^n) = n \ln(a)\)
• Quotient Rule: \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\)

These properties allow for breaking down complex logarithmic functions into more manageable parts. This simplification process can be particularly beneficial in algebraic manipulation, calculus operations, and signal processing. By understanding and applying these rules appropriately, one can solve equations and understand graphs more efficiently.