Problem 19 - 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 \(x, y > 0\). $$\log \frac{\sqrt[4]{x}}{y^{-1}}$$

Step-by-Step Solution

Answer

The simplified expression of \( \log \frac{\sqrt[4]{x}}{y^{-1}} \) is \( \frac{1}{4}\log x +\log y \)

Step 1: Apply the Logarithm of a Quotient Property

This property asserts that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Hence, we can write \( \log \frac{\sqrt[4]{x}}{y^{-1}} \) as \( \log\sqrt[4]{x}-\log y^{-1} \).

Step 2: Express Radicals as Powers

By understanding that a root can be rewritten as a power. In this case specifically, the fourth root of \(x\), can be written as \(x^{(1/4)}\). Thus, the expression now becomes \( \log x^{(1/4)}-\log y^{-1} \).

Step 3: Apply Logarithm of a Power Property

The logarithm of a power property dictates that the logarithm of a number raised to a power equals the product of the power and the logarithm of the number. This can be applied to both terms in our expression. We then get \( (1/4)*\log x -(-1)*\log y \), which can be simplified to \( \frac{1}{4}\log x +\log y \).

Key Concepts

Logarithm Properties

Logarithms turn multiplication into addition and division into subtraction. These transformations make them very useful for simplifying complex expressions. One key property is the logarithm of a quotient, which says that when you take the logarithm of a fraction, you can express it as the difference between the logarithm of the numerator and the logarithm of the denominator. For instance:

• The expression \( \log \frac{a}{b} \) can be rewritten as \( \log a - \log b \).

Additionally, the logarithm of a power property states that for any positive number \( a \) and any real number \( n \), the logarithm of \( a^n \) can be simplified to \( n \cdot \log a \).

• In other words, \( \log(a^n) = n \cdot \log a \).

This is particularly helpful for converting expressions with exponents, making them more manageable to work with. By applying these properties, one can simplify expressions involving logs considerably, as seen in the solution provided.

Radicals

Radicals often appear as roots in mathematical expressions. They can sometimes make problems difficult to solve. To ease the complexity of these problems, you can rewrite radicals as powers or exponents. This transformation makes it simpler to apply properties of exponents and logarithms.
For example, a square root, like \( \sqrt{a} \), can be written as \( a^{1/2} \). Similarly, a fourth root, such as \( \sqrt[4]{x} \), is equivalent to \( x^{1/4} \).

• Rewriting roots as exponents is crucial for simplifying expressions within log problems.

This allows you to apply logarithm properties more effectively to solve or simplify expressions.

Exponents

Exponents signify repeated multiplication of a base number. They play a key role in simplifying mathematical problems involving logarithms and radicals. When a number is raised to an exponent, it is essentially a short form of multiplying the number by itself several times.
Key exponent properties can make manipulating expressions smoother:

• Product Rule: \( a^m \times a^n = a^{m+n} \)
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power Rule: \((a^m)^n = a^{m \cdot n}\)

By integrating these properties, it becomes much easier to approach and simplify problems, especially when they are paired with properties of logarithms. Changing radicals into their exponential form not only simplifies the problem but also aligns easily within the realm of logarithmic functions, creating a seamless process for problem-solving.