Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify the given expression. $$ e^{\ln x^{2}-y \ln x} $$

Step-by-Step Solution

Answer

\( x^{2-y} \)

Step 1: Simplify Inside the Exponential

The expression inside the exponential is \( \ln x^{2} - y \ln x \). Rewrite \( \ln x^{2} \) as \( 2 \ln x \) using the logarithm power rule. Thus, the expression becomes \( 2 \ln x - y \ln x \).

Step 2: Factor Out Common Terms

Notice that \( 2 \ln x - y \ln x \) can be factored to \((2-y) \ln x\) since \( \ln x \) is common in both terms.

Step 3: Use Exponential and Logarithm Properties

Recall that \( e^{a \ln b} = b^{a} \). Apply this property to \( e^{(2-y) \ln x} \), simplifying it to \( x^{2-y} \).

Key Concepts

Logarithmic Properties

Understanding logarithmic properties can simplify complex mathematical expressions. The power rule in logarithms states

• If you have an expression like \( \ln x^a \), it can be rewritten as \( a \ln x \). This transformation is due to the power rule: \( \ln(x^a) = a \cdot \ln(x) \).
• It allows movement of the exponent to the front of the logarithm helping simplify expressions.

In our original problem, \( \ln x^2 \) is simplified to \( 2 \ln x \) by applying this rule. Recognizing when to apply these logarithmic properties is key to tackling problems involving exponents and logarithms. Knowing these rules makes it easier to rewrite and simplify expressions.

Simplifying Expressions

Simplifying expressions is about finding ways to express them in a cleaner, reduced form without changing their value. This often involves recognizing patterns and applying mathematical rules. When working with exponential and logarithmic expressions, simplification might include:

• Combining like terms: Notice in the expression \( 2 \ln x - y \ln x \), the term \( \ln x \) appears in both parts, allowing us to combine them into \( (2-y) \ln x \).
• Applying mathematical identities: Once you have a simplified expression inside a logarithm or exponent, you can often simplify further by applying properties like factoring common terms.

By practicing these steps, you can make expressions easier to work with and understand, reducing the complexity significantly.

Algebraic Manipulation

Algebraic manipulation involves using algebraic rules and operations to rearrange and simplify equations and expressions. In exponential and logarithmic contexts:

• You may encounter situations where you can use properties such as \( e^{a \ln b} = b^a \), which allows for the transformation of exponential expressions involving logarithms into a more digestible form.
• In the example, we applied this property to \( e^{(2-y) \ln x} \) which led to simplifying the expression to \( x^{2-y} \).
• Understanding such algebraic transformations can aid in simplifying intricate expressions to forms that are easier to interpret and use in calculations.

By mastering these techniques, you become adept at tweaking and reorganizing algebraic expressions to solve problems more efficiently.