Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\ln \frac{x}{\sqrt{x^{2}+1}}$$.

Step-by-Step Solution

Answer

The expanded expression using the properties of logarithms is \(\ln(x) - \frac{1}{2}\ln(x^{2}+1)\)

Step 1: Apply Quotient Rule

Start by applying the quotient rule to the expression, which will split the logarithm into two: \(\ln(x) - \ln(\sqrt{x^{2}+1})\).

Step 2: Apply Power Rule

Next, apply the power rule to the square root in the second logarithmic term: \(\ln(x) - \frac{1}{2}\ln(x^{2}+1)\).

Step 3: Simplify the Expression

Finally, check if there are any further simplifications that can be made. In this instance, \(\ln(x) - \frac{1}{2}\ln(x^{2}+1)\) is as simplified as it can be.

Key Concepts

Understanding the Logarithmic Quotient Rule

The logarithmic quotient rule is a powerful tool for breaking down complex logarithmic expressions into simpler components. It specifically states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. This is mathematically expressed as: \[\begin{equation}\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\end{equation}\]Here, \(b\) represents the base of the logarithm, while \(M\) and \(N\) are the numerator and denominator of the quotient, respectively.
This rule is a reflection of the corresponding rule for exponents, that is, \(b^{\log_b(M) - \log_b(N)} = \frac{b^{\log_b(M)}}{b^{\log_b(N)}} = \frac{M}{N}\). The quotient rule simplifies the process of expanding logarithmic expressions, which can be especially useful when dealing with variables and complex functions, as seen in the original exercise.

Applying the Logarithmic Power Rule

Diving into the logarithmic power rule, we can understand it as an extension of the fundamental nature of logarithms and exponents. This rule allows us to move the exponent in a logarithmic argument to the front, as a multiplicative factor. Mathematically, it is shown as:\[\begin{equation}\log_b(M^p) = p \cdot \log_b(M)\end{equation}\]where \(p\) is the exponent and can be any real number.

This rule greatly simplifies the process of expanding logarithmic expressions that involve exponential terms. For example, when you encounter a square root within a logarithm, you can treat it as the original expression raised to the half power. Applied in the context of the given exercise, the power rule transforms the logarithm of \(\sqrt{x^{2}+1}\) into \(\frac{1}{2}\log(x^{2}+1)\), which simplifies the expression and makes it easier to manage and understand.

Simplifying Logarithmic Expressions

When simplifying logarithmic expressions, it's not only about knowing the rules but also about understanding when the expression is in its simplest form. To simplify logarithms, you can use a variety of properties, including the product rule, quotient rule, and power rule. Alongside these, recognizing when a logarithm cannot be further simplified is crucial. This often results in leaving the expression with addition or subtraction of logarithmic terms, possibly with coefficients, as they cannot be combined in the same way that like terms in polynomial expressions can.

In the step by step solution of the exercise, after applying both the quotient and power rules, we reach a point where no more simplifications can be made. This leaves us with \(\ln(x) - \frac{1}{2}\ln(x^{2}+1)\), which is the expression's most reduced form given the lack of like terms or further exponent manipulation. It's essential for students to practice these properties to gain a deeper intuition for recognizing the simplest form of logarithmic expressions.