Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

\(\ln e^{x^{2}} = x^{2}\)

Step 1: Identify the Logarithm Property

First, point out the corresponding property of natural logarithms that will assist in simplifying this expression. Recognize that for any real number x, \(\ln e^{x} = x\).

Step 2: Apply the Logarithm Property

Now, apply the logarithm property to simplify the given expression. Here, x is actually \(x^2\). Thus, reduce \(\ln e^{x^{2}}\) to \(x^{2}\) according to the property of logarithms.

Step 3: Final Simplified Expression

So, after simplifying the given expression, it's seen that \(\ln e^{x^{2}} = x^{2}\). This is the simplified form of the provided logarithmic expression.

Key Concepts

Natural Logarithm

When you deal with logarithms, you will often come across the term "natural logarithm." This refers to logarithms with a specific base, the mathematical constant \(e\), which is approximately 2.71828. This special kind of log is denoted as \(\ln\), which stands for "logarithm naturalis," reflecting its Latin origin.

Natural logarithms serve numerous purposes in both mathematics and the real world, from integrating certain functions in calculus to calculating time in financial models.

Logarithms eliminate complexities by transforming multiplications into additions and powers into multiplications. This feature simplifies solving problems involving exponential growth or decay, which are common in real-life applications such as population growth models and radioactive decay.

• The expression \(\ln (e) = 1\) simplifies calculations since \(e^1 = e\).
• Its property \(\ln (e^x) = x\) turns exponents into simple linear expressions.

Logarithm Properties

Understanding the properties of logarithms is essential when working with logarithmic expressions. These properties help in simplifying complex expressions and solving equations efficiently.

One key property is the Power Rule. This rule states that \(\ln (a^b) = b \cdot \ln (a)\). It allows you to move the exponent in front as a multiplier. For example, in the expression \(\ln e^{x^2}\), \(x^2\) can be moved before the logarithm, resulting in \(x^2 \cdot \ln (e)\).

• Addition and Subtraction Rule: \(\ln (a \cdot b) = \ln a + \ln b\), and \(\ln \frac{a}{b} = \ln a - \ln b\).
• Change of Base Formula: Convert logs from one base to another, \(\log_b a = \frac{\ln a}{\ln b}\).

Each applies these rules to transform and manipulate logarithmic expressions for simplification or solution.

Simplifying Expressions

Simplifying expressions, particularly those involving logarithms, combines the use of logarithmic properties and algebraic techniques. Keeping these strategies in mind enhances your ability to solve problems efficiently.

For the expression \(\ln e^{x^2}\), simplifying it is achieved by directly applying the Power Rule property. Since \(\ln (e) = 1\), any expression \(\ln (e^x)\) equals \(x\). Therefore, \(\ln e^{x^2}\) simplifies to just \(x^2\).

• Check for applicable properties such as the Power Rule or other basic identities, like \(\ln 1 = 0\).
• Look for opportunities to utilize inverse operations that can cancel out parts of the expression.

Simplification helps to break down complex problems into manageable pieces, leading to clearer solutions and more straightforward calculations.