Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify the given expression. $$ e^{\ln 3+2 \ln x} $$

Step-by-Step Solution

Answer

The simplified expression is \( 3x^2 \).

Step 1: Use Logarithm Properties

Recall the property of logarithms: \( a \ln b = \ln(b^a) \). Applying this to the term \( 2 \ln x \), we get \( \ln(x^2) \). Thus, the expression becomes: \( e^{\ln 3 + \ln(x^2)} \).

Step 2: Combine Logarithms

Using the logarithm addition rule, \( \ln a + \ln b = \ln(ab) \), combine the logarithms: \( \ln 3 + \ln(x^2) = \ln(3x^2) \). The expression is now \( e^{\ln(3x^2)} \).

Step 3: Simplify Exponential and Logarithm

Apply the property \( e^{\ln a} = a \) to simplify the expression: \( e^{\ln(3x^2)} = 3x^2 \).

Key Concepts

Understanding Logarithm Properties

Logarithms can be quite confusing at first, but they follow some straightforward rules. One of these is the property: \( a \ln b = \ln(b^a) \). This means you can move a multiplication inside a logarithm to an exponent. For instance, \( 2 \ln x \) transforms into \( \ln(x^2) \).
Logarithms also have the addition rule: \( \ln a + \ln b = \ln(ab) \). This means when you add two logs together, you can transform it into a single log by multiplying the inside values. For example, combining \( \ln 3 + \ln(x^2) \) results in \( \ln(3x^2) \).

• Use these properties to manipulate and simplify logarithmic expressions.
• Practice moving coefficients in and out of logarithms.
• Get comfortable with combining multiple logarithmic terms into one.

Demystifying Exponential Functions

Exponential functions often involve the natural number \( e \), which is approximately 2.718. In expressions like \( e^{\ln a} \), the base \( e \) and the natural log (\( \ln \)) essentially cancel each other out, simplifying the expression to \( a \). This emerges from the inverse relationship between exponentials and logarithms.
For example, in our exercise, \( e^{\ln(3x^2)} \) simplifies directly to \( 3x^2 \). Understanding exponential functions involves recognizing that the exponent tells you how many times to use the base in a multiplication.

• Recognize that \( e \) and \( \ln \) are inverse operations.
• Recognize common bases in exponential expressions, especially \( e \).
• Apply the exponential and logarithmic properties to simplify expressions.

Steps to Simplifying Expressions

The process of simplifying expressions can appear tricky, but following a few key strategies can make it more manageable. Let’s apply these to our expression \( e^{\ln 3 + 2 \ln x} \).
1. First, apply logarithm properties to simplify individual terms.
2. Combine the logs using their respective rules, like we did by getting \( \ln(3x^2) \).
3. Use properties of exponentials and logs to simplify further. Exponential and logarithm hardly coexist without simplifications. In our expression, because of \( e \) raised to a log, it simplifies exactly to the value inside the log, which resulted in \( 3x^2 \).

• Identify which properties apply to different parts of the expression.
• Always look for patterns and relationships between terms, like recognizing inverse operations.
• Break down the problem into smaller parts and operate on each part, combining at the final step.