Problem 21 - College Algebra Essentials

Question

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{b}\left(x^{2} y\right)\)

Step-by-Step Solution

Answer

The expanded form of the logarithmic expression \( \log _{b}\left({x^{2} y}\right) \) is \( 2 \cdot \log_b x + \log_b y \).

Step 1: Apply the Product Rule

The given logarithmic expression \( \log _{b}\left({x^{2} y}\right) \) can be seen as the log of two values being multiplied (with \(M = x^2\) and \(N = y^)). Therefore, we can use the Product Rule of Logarithms to separate this into two terms. This gives \( \log_b (x^2) + \log_b y \).

Step 2: Apply the Power Rule

In the term \( \log_b (x^2) \), \(x^2\) can be seen as \(x\) raised to the power 2. Therefore, we can use the Power Rule of Logarithms to bring the exponent out in front as a multiplier. This gives \( 2 \cdot \log_b x + \log_b y \).

Step 3: Final Expanded Expression

The final expanded form of the logarithmic expression is \( 2 \cdot \log_b x + \log_b y \). If the values of \(b\), \(x\), and \(y\) are known and can be evaluated without a calculator, this can be further simplified. Otherwise, this is the most simplified form of the given expression.

Key Concepts

Product rule of logarithms

The product rule of logarithms is an essential property that helps us break down complex logarithmic expressions. When you have a logarithm of a product, such as \( \log_b (MN) \), where \( M \) and \( N \) are numbers or expressions, you can express it as the sum of the logarithms of the individual factors.
This means \( \log_b (MN) = \log_b M + \log_b N \).
This property is particularly useful for expanding expressions into simpler components.

• It simplifies multiplication inside the log into an addition outside.
• By doing this, it becomes easier to deal with each term separately, especially if there are additional properties that can be applied to individual terms.

In practice, whenever you see a logarithm of multiplied terms, remember to use the product rule to split them apart into a sum.
This makes further simplifications and calculations much more manageable.

Power rule of logarithms

The power rule of logarithms allows us to simplify expressions where a term inside a logarithm is raised to an exponent.
When you encounter something like \( \log_b (M^n) \), you can apply the power rule, which states \( \log_b (M^n) = n \cdot \log_b M \).
This effectively moves the exponent in front of the logarithm as a multiplier, turning the problem of dealing with powers into a straightforward multiplication problem.

• This rule is helpful because it can simplify logarithmic expressions significantly.
• It converts the operation of exponentiation into a multiplication outside the log, making handling the expression easier.

When you see terms in logarithmic expressions with exponents, apply the power rule to bring the exponent down front.
This helps simplify and expand expressions, paving the way for further reduction if other logarithms rules can be employed.

Expanding logarithmic expressions

Expanding logarithmic expressions involves using properties like the product and power rules to rewrite expressions in a longer yet simpler form.
The goal is to break down a complicated log expression into its constituent parts so it's easier to work with, understand, and possibly evaluate.
For example, given \( \log_b (x^2 y) \), you'd first use the product rule to separate it into \( \log_b (x^2) + \log_b y \).
The expression can be expanded further using the power rule, resulting in \( 2 \cdot \log_b x + \log_b y \).

• Expanding doesn't necessarily solve the expression entirely but transforms it into a format that's often more intuitive to further simplify or calculate.
• By expanding expressions, you often reveal hidden relationships and easier paths to solution through simplification.

The expanded form is often the first step towards further simplification, evaluation, or integration into larger algebraic calculations or problems.