Problem 20 - Precalculus : Building Concepts and Connections

Question

In Exercises \(15-20,\) use the properties of logarithms to simplify each expression by eliminating all exponents and radicals. Assume that \(x, y > 0\). $$\log \frac{\sqrt[3]{x}}{y^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{3} \log(x) - 2\log(y)\).

Step 1: Identify the components of the expression

The expression \(\log \frac{\sqrt[3]{x}}{y^{2}}\) is comprised of a logarithm with a fraction in its base. The numerator is a cube root \(\sqrt[3]{x}\) and the denominator is an exponent \(y^{2}\).

Step 2: Apply the quotient rule for logarithms

The quotient rule states that the logarithm of a fraction equals the difference of the logarithms of its parts. This allows us to split our expression into \(\log(\sqrt[3]{x}) - \log(y^{2})\).

Step 3: Apply the power rule for logarithms

The power rule states that logarithm of a positive number to a given power is the same as the product of that power and the logarithm of the positive number. So, \(\log(\sqrt[3]{x})\) becomes \(\frac{1}{3} \log(x)\) and \(\log(y^{2})\) becomes \(2\log(y)\). Thus, the simplified expression is: \(\frac{1}{3} \log(x) - 2\log(y)\).

Key Concepts

Logarithm Simplification

Logarithm simplification is all about making a complex logarithmic expression easier to understand and work with. This process involves using logarithmic properties to break down larger expressions into simpler components.
By simplifying, you can make solve and understand problems quicker, especially in algebra and calculus.
In our exercise, we start with a logarithm of a fraction:

• Numerator: Cube root of \(x\), which is \(\sqrt[3]{x}\).
• Denominator: \(y^2\).

The goal is to break this down into a more manageable form using the properties of logarithms.

Quotient Rule for Logarithms

The quotient rule for logarithms is a handy tool when dealing with fractions inside a log. It simplifies a logarithm of a division into the difference between two logs.
Here is the rule:

• For any positive numbers \(a\) and \(b\), \(\log \left( \frac{a}{b} \right) = \log(a) - \log(b)\).

This rule kicks in perfectly for our exercise because our base is a fraction, \(\frac{\sqrt[3]{x}}{y^2}\).
This expression can be split as follows:

• \(\log(\sqrt[3]{x}) - \log(y^2)\)

Now, both parts are ready for further simplification using other rules, like the power rule.

Power Rule for Logarithms

The power rule is another key property that helps in simplifying logarithms involving exponents or roots.
It states the following:

• For any positive number \(a\) and real number \(b\), \(\log(a^b) = b\log(a)\).

It can be applied both to exponents and roots by expressing the root as an exponent. Specifically:

• \(\log(\sqrt[3]{x})\) becomes \(\frac{1}{3}\log(x)\) because \(\sqrt[3]{x}\) is \(x^{1/3}\).
• \(\log(y^2)\) becomes \(2\log(y)\) since the power is 2.

So, using the power rule, the simplified result from our expression is \(\frac{1}{3} \log(x) - 2\log(y)\).
This form is much cleaner and easier to interpret than the original.