Problem 19 - College Algebra

Question

For the following exercises, condense to a single logarithm if possible. $$ \log \left(x^{2} y^{3} \sqrt[3]{x^{2} y^{5}}\right) $$

Step-by-Step Solution

Answer

\(\log(x^{8/3} y^{14/3})\)

Step 1: Apply Properties of Exponents and Radicals

Identify and simplify the radicals. The expression inside the logarithm is \(x^{2} y^{3} imes (x^{2} y^{5})^{1/3}\). Apply the property of exponents \((a^m)^n = a^{m imes n}\) which simplifies the radicals to \(x^{2/3} y^{5/3}\).

Step 2: Combine Exponents

Multiply the expression \(x^{2} y^{3}\) by the simplified expression \(x^{2/3} y^{5/3}\). Using the property \(a^m imes a^n = a^{m+n}\), combine the powers for each base:- For base \(x\): \(x^2 imes x^{2/3} = x^{2 + 2/3} = x^{8/3}\).- For base \(y\): \(y^3 imes y^{5/3} = y^{3 + 5/3} = y^{14/3}\).

Step 3: Express as a Single Logarithm

The condensed expression is now \( \log(x^{8/3} y^{14/3}) \). Apply the property for logarithms \(\log(a^m b^n) = m \log a + n \log b\) backwards to recognize that this is already a single logarithm of a product.

Key Concepts

Properties of Exponents

Exponents are a crucial tool in algebra and understanding their properties can simplify many mathematical expressions.
A fundamental property is when an exponent is raised to another power like \[(a^m)^n = a^{m \times n}\].
This allows us to handle more complex expressions by breaking them into simpler parts.
For example, in our exercise, the expression \((x^2 y^3) (x^2 y^5)^{1/3}\) includes radicals that can be managed using exponent rules.
We convert \((x^2 y^5)^{1/3}\) into exponents as \(x^{2/3} y^{5/3}\) by applying the formula above.
When dealing with products of powers with the same base, remember:

• Multiply: \(a^m \times a^n = a^{m+n}\)
• For example, \(x^2 \times x^{2/3} = x^{8/3}\)
• Similarly with y: \(y^3 \times y^{5/3} = y^{14/3}\)

Mastering these simple patterns helps to eliminate radicals and streamline calculations, enhancing our ability to simplify expressions elegantly.

Radicals

Radicals, often seen as square roots, cube roots and beyond, represent fractional exponents in algebra.
The cube root of a number like \(a\) is expressed as \(a^{1/3}\).
This link between radicals and exponents is fundamental for rearranging and simplifying expressions.
Consider the expression \((x^2 y^5)^{1/3}\), which is inside our original logarithm.
Converting this to fractional exponents gives:

• \(x^{2/3}\) because the power 2 is divided by 3
• \(y^{5/3}\) similarly, the power 5 divided by 3

This conversion allows the use of properties of exponents to further simplify and combine terms, making calculations and algebraic manipulations consistent and manageable.
By mastering how radicals work as fractional exponents, we facilitate easier handling of complex structures.

Combining Logarithms

Combining logarithmic expressions into a single entity is a powerful technique that uses logarithmic properties.
One key property is that the logarithm of a product can be expressed as the sum of logarithms: \(\log(a \cdot b) = \log a + \log b\).
However, the opposite is true when simplifying: turning sums back into a single logarithm expression.
In our task, we began with a log expression: \(\log (x^{8/3} y^{14/3})\), representing a simplified product.
This is the product of two powers, and thus already a single logarithm of a product, not separate logs.
To clarify, if we had separate logs which add, like \(\log a + \log b\),
we could condense that into \(\log(ab)\).
Utilizing these properties allows the effective simplification and combination of logarithmic expressions, giving cleaner, clearer results.