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Chapter 2: Problem 9

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

Short Answer

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

Step by step solution

01

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: \( e^{\ln 3 + \ln(x^2)} \).
02

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 \( e^{\ln(3x^2)} \).
03

Simplify Exponential and Logarithm

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

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