Problem 69 - Precalculus with Calculus Previews

Question

In Problems \(69-72\), use the laws of logarithms in Theorem \(6.2 .1\) so that \(\ln y\) contains no products, quotients, or powers. $$ y=\frac{x^{10} \sqrt{x^{2}+5}}{\sqrt[3]{8 x^{3}+2}} $$

Step-by-Step Solution

Answer

\( \ln y = 10 \ln x + \frac{1}{2} \ln(x^2 + 5) - \frac{1}{3} \ln(8x^3 + 2) \)

Step 1: Express the Equation in Logarithmic Form

First, take the natural logarithm of both sides. We have the equation \( y = \frac{x^{10} \sqrt{x^2 + 5}}{\sqrt[3]{8x^3 + 2}} \). Taking the natural log of both sides gives us: \( \ln y = \ln \left( \frac{x^{10} \sqrt{x^2 + 5}}{\sqrt[3]{8x^3 + 2}} \right) \).

Step 2: Apply the Quotient Rule of Logarithms

Using the logarithm quotient rule, \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \), we can rewrite the expression as: \( \ln y = \ln(x^{10} \sqrt{x^2 + 5}) - \ln(\sqrt[3]{8x^3 + 2}) \).

Step 3: Apply the Product Rule of Logarithms

Using the logarithm product rule, \( \ln(ab) = \ln a + \ln b \), decompose \( \ln(x^{10} \sqrt{x^2 + 5}) \) into two separate logarithms: \( \ln(x^{10}) + \ln(\sqrt{x^2 + 5}) \). Thus, the equation becomes: \( \ln y = \ln(x^{10}) + \ln(\sqrt{x^2 + 5}) - \ln(\sqrt[3]{8x^3 + 2}) \).

Step 4: Apply the Power Rule of Logarithms

Use the power rule, \( \ln(a^b) = b\ln a \), to simplify each term: - For \( \ln(x^{10}) \), we get \( 10 \ln x \). - For \( \ln(\sqrt{x^2 + 5}) \) \((where \( \sqrt{x^2 + 5} = (x^2 + 5)^{1/2} \))\), we get \( \frac{1}{2}\ln(x^2 + 5) \). - For \( \ln(\sqrt[3]{8x^3 + 2}) \) \((where \( \sqrt[3]{8x^3 + 2} = (8x^3 + 2)^{1/3} \))\), we get \( \frac{1}{3}\ln(8x^3 + 2) \).

Step 5: Final Expression for \(\ln y\)

Put all the terms back together in the simplified form so that no products, quotients, or powers remain inside the logarithms: \( \ln y = 10 \ln x + \frac{1}{2} \ln(x^2 + 5) - \frac{1}{3} \ln(8x^3 + 2) \). This is the expression for \( \ln y \) without products, quotients, or powers inside the logarithms.

Key Concepts

Logarithm Quotient Rule

The Logarithm Quotient Rule is an essential tool when working with logarithms. It helps simplify the logarithm of a division between two quantities. The rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator.

Mathematically, this is expressed as:

• \( \ln \left(\frac{a}{b}\right) = \ln a - \ln b \)

In the original exercise, we applied this rule to express the natural logarithm of a large quotient as the difference of two separate logarithms.

By breaking down the quotient into smaller parts, each represented by its own logarithm, we're able to further simplify the expression. This method helps us manage more complex logarithmic expressions by creating simpler components that we can individually manipulate.

This foundational concept is not only applicable in algebra but also in calculus and other higher-level mathematical fields, providing a versatile technique for dealing with ratios within logarithms.

Logarithm Product Rule

The Logarithm Product Rule is a powerful method for dealing with logarithms involving multiplication. This rule states that the logarithm of a product is the sum of the logarithms of the individual factors.

In mathematical terms, the rule is:

• \( \ln(ab) = \ln a + \ln b \)

During the exercise, we used this rule to split the logarithm of a product inside the original expression. Specifically, we decomposed \( \ln(x^{10} \sqrt{x^2 + 5}) \) into its components, allowing us to treat each factor separately.

Breaking down a larger product into smaller, more manageable terms can significantly simplify the calculations.

This rule is especially useful in simplifying expressions before applying more specific rules such as the logarithm power rule. The clarity it brings to complex equations makes it an essential tool in simplifying logarithmic expressions across various branches of mathematics.

Logarithm Power Rule

The Logarithm Power Rule is an efficient strategy to handle powers within logarithmic functions. The rule enables us to transform the exponent of a number into a coefficient, thereby simplifying the logarithmic expression.

The rule can be represented mathematically as:

• \( \ln(a^b) = b \ln a \)

In our solution, we applied this rule to each term that contained a power. This included rewriting \( \ln(x^{10}) \) as \( 10 \ln x \), bringing down the exponent as a multiplying factor.

We also used it on roots by recognizing them as expressions with fractional exponents:

• \( \ln(\sqrt{x^2 + 5}) = \ln((x^2 + 5)^{1/2}) = \frac{1}{2}\ln(x^2 + 5) \)
• \( \ln(\sqrt[3]{8x^3 + 2}) = \ln((8x^3 + 2)^{1/3}) = \frac{1}{3}\ln(8x^3 + 2) \)

Applying this rule helps remove complexities related to powers within logarithmic terms, leading to a simpler form.

The ability to transform exponents into coefficients allows mathematicians to further manipulate and solve logarithmic expressions, making the power rule a crucial aspect of logarithmic algebra.