Problem 44 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$\sqrt{2} \ln e^{\sqrt{2}}$$

Step-by-Step Solution

Answer

The expression simplifies to 2.

Step 1: Understanding the Logarithmic Identity

Remember that the natural logarithm \( \ln \) has the property \( \ln(a^b) = b \ln a \). Also note that \( \ln e = 1 \) because \( e \) is the base of the natural logarithm.

Step 2: Applying the Logarithmic Identity

In the expression \( \ln e^{\sqrt{2}} \), apply the identity to rewrite it as \( \sqrt{2} \ln e \). Since \( \ln e = 1 \), it simplifies further to \( \sqrt{2} \cdot 1 = \sqrt{2} \).

Step 3: Simplifying the Entire Expression

Now, substitute the result from Step 2 back into the original expression \( \sqrt{2} \ln e^{\sqrt{2}} \), which now becomes \( \sqrt{2} \cdot \sqrt{2} \).

Step 4: Multiplying the Square Roots

When you multiply \( \sqrt{2} \times \sqrt{2} \), it simplifies to \( 2 \), because \( \sqrt{2} \times \sqrt{2} = 2 \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with the base \( e \), where \( e \approx 2.71828 \). It is widely used in mathematics, especially in calculus and exponential growth models. The main characteristic of the natural logarithm is its relationship with the exponential function. The function \( \ln(e^x) = x \) is what makes it particularly useful.- **Why use natural logarithms?**
They simplify complex mathematical expressions involving exponential functions.
- **Fun fact:** The natural logarithm can be expressed as an integral: \( \ln(x) = \int_1^x \frac{1}{t} \, dt \).In solving logarithmic problems, remembering that \( \ln(e) = 1 \) is quite crucial. It simplifies expressions significantly, as seen in the problem, where \( \ln(e^{\sqrt{2}}) \) uses these properties to break down into simpler components.

Logarithmic Identity

Logarithmic identities are handy shortcuts to simplify complex logarithmic expressions. These identities stem from the fundamental properties of logarithms.- The key identity in this context is: \( \ln(a^b) = b \ln a \).
This identity stems from exponent rules and allows the exponents to "come down" out of the logarithm, which is useful for simplification.- **How it works:**
In the problem, we have \( \ln(e^{\sqrt{2}}) \). Using the identity, this becomes \( \sqrt{2} \cdot \ln(e) \). Given \( \ln(e) = 1 \), it simplifies directly to \( \sqrt{2} \).Understanding and applying these identities can transform and ease the computation process, especially when evaluating expressions manually.

Square Roots

Square roots are numbers that, when multiplied by themselves, give the original number. The square root of a number \( x \) is often expressed as \( \sqrt{x} \).- **Why they matter:**
In mathematics, square roots appear in many different contexts, including geometry, algebra, and calculus. They are essential when solving quadratic equations or working with areas and volumes.- **Simplifying square roots:**
When you multiply two identical square roots, the result is the original number: \( \sqrt{a} \times \sqrt{a} = a \).
This principle is seen when \( \sqrt{2} \times \sqrt{2} = 2 \) in the given problem.Square roots are straightforward yet powerful tools for simplifying expressions, especially when combined with logarithms and other mathematical operations.