Problem 24 - Precalculus : Building Concepts and Connections

Question

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\ln \frac{\sqrt[4]{y^{3}}}{e^{5}}$$

Step-by-Step Solution

Answer

The simplified logarithmic expression is \(\frac{3}{4}\) \(ln(y)\) - 5.

Step 1: Apply the Quotient Rule

The quotient rule in logarithms says that the logarithm of a quotient can be written as the difference of the logarithms. So, the logarithm expression is broken down into difference of two logarithms as follows: \(ln(\frac{\sqrt[4]{y^{3}}}{e^{5}})\) = \(ln(\sqrt[4]{y^{3}})\) - \(ln(e^{5})\).

Step 2: Evaluate the Logarithm of Base e

Note that the natural logarithm \(ln(e^{5})\) is just 5, since the natural logarithm has a base of e. Therefore, the expression can be further simplified as follows: \(ln(\sqrt[4]{y^{3}})\) - 5.

Step 3: Apply the Rule of Logarithm of a Root

The rule of logarithm of a root says that the logarithm of a root can be written as a fraction. Let's apply this rule to simplify the term \(ln(\sqrt[4]{y^{3}})\). We get: \(\frac{3}{4}\) \(ln(y)\) - 5.

Key Concepts

Quotient Rule

The quotient rule in logarithms is a handy tool that helps us break down complex expressions. When you have a logarithm of a quotient, such as \( \ln \left( \frac{A}{B} \right) \), it can be simplified to \( \ln(A) - \ln(B) \). This conversion makes it easier to handle and solve logarithmic expressions. In practical terms:

• It simplifies the evaluation of logarithmic expressions.
• Helps identify parts of the expression that can be solved individually.
• Makes complex expressions more manageable by separating them into simpler components.

In the given exercise, we used the quotient rule to simplify \( \ln \left( \frac{\sqrt[4]{y^{3}}}{e^{5}} \right) \) to \( \ln(\sqrt[4]{y^{3}}) - \ln(e^{5}) \). Splitting complex fractions into differences makes the next steps much clearer.

Natural Logarithm

A natural logarithm, denoted as \( \ln(x) \), has a special base known as 'e', which is approximately 2.718. The main purpose of natural logarithms is to simplify expressions where this base is involved.Understanding the properties:

• The inverse of natural logarithms is the exponential function \( e^x \).
• With base 'e', \( \ln(e^n) \) is simplified directly to 'n'.
• It appears frequently in calculus and applied mathematics because of its natural growth patterns.

In our problem, \( \ln(e^{5}) \) simplifies to 5 because the effect of the base 'e' and the logarithm cancel each other out entirely. This property made it easy for us to reduce the expression to \( \ln(\sqrt[4]{y^{3}}) - 5 \).

Logarithm of a Root

When dealing with logarithms of roots, understanding and applying the rule for roots helps simplify expressions substantially. The rule states that the logarithm of a root, such as \( \sqrt[n]{A} \), can be expressed as a fraction: \( \frac{1}{n} \ln(A) \).Key points to remember:

• Converting roots into fractional exponents aids in simplification.
• Allows for the transformation of complex numbers into more easily handled terms.
• This principle holds whether the root is square, cube, fourth, or any other kind.

In the example, \( \ln(\sqrt[4]{y^{3}}) \) was rewritten as \( \frac{3}{4} \ln(y) \), because the exponent 3 on \/( quantity \( y \) is multiplied by the fraction \( \frac{1}{4} \) from the fourth root. Applying this rule enables us to further simplify and evaluate very complex-looking logarithms.