Problem 15 - 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 \(x, y > 0\). $$\log \left(x y^{3}\right)$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\log (x y^3)\) is \(\log x + 3 \log y\)

Step 1: Apply logarithmic distributive property

Utilize the property \(\log_a (mn) = \log_a m + \log_a n\) to break up the logarithm of a product into the sum of two logarithms. This gives: \(\log (x y^3) = \log x + \log y^3\)

Step 2: Apply power property of logarithms

Applying the property \(\log_b (m^n) = n \log_b m\), allows us to move the exponent 3 in front of the logarithm. Hence: \(\log x + \log y^3 = \log x + 3 \log y\)

Key Concepts

Product Property of Logarithms

When faced with a logarithm that contains a product inside, such as \( \log(xy^3) \), you can use the product property of logarithms to simplify it. This property states:\[\log_a (mn) = \log_a m + \log_a n\]

• The logarithm of a product is equal to the sum of the logarithms of the individual factors.
• In our exercise, this means that \( \log(xy^3) \) can be broken down into \( \log x + \log y^3 \).

Rather than dealing with complex expressions as a whole, breaking them down using this property makes them simpler to manage.Break down each component, so you can more easily use other logarithmic properties to further simplify the expression.This property highlights how logarithms convert multiplication operations into addition, making the expression easier to work with.

Power Property of Logarithms

Another important property to consider is the power property of logarithms, which is helpful when you have expressions with exponents, like \( y^3 \) in our problem. This property is expressed as:\[\log_b (m^n) = n \log_b m\]

• It allows you to move the exponent in front of the logarithm as a coefficient.
• So, in our example, \( \log y^3 \) becomes \( 3 \log y \).

This simplifies equations by turning multiplication into scalar multiplication before the logarithm.By bringing the exponent down, you can more easily add or manage the terms in the expression. For scenarios involving powers, this property is key to streamlining complex logarithmic elements.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves applying properties like the product and power properties consecutively. Doing so transforms logarithmic expressions into more workable formats.

• Start by breaking down complex products using the product property. This turns the logarithm of a product into a sum of logarithms. For example, \( \log(xy^3) \) becomes \( \log x + \log y^3 \).
• Next, apply the power property to deal with exponents in the terms. Take \( \log y^3 \), and simplify it to \( 3 \log y \).

These steps eliminate exponents and convert multiplications inside the logarithm into additions, facilitating a clearer calculation process.Understanding and applying these properties not only simplifies expressions but also aids in solving logarithmic equations efficiently.Through practice, you'll gain confidence in transforming even intricate logarithmic statements using these foundational tools.