Problem 44 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

2

Step 1: Simplify the Expression Inside the Logarithm

We begin by evaluating the expression inside the logarithm: \[ e^{\sqrt{2}} \]The natural logarithm of something is simply the power to which the base 'e' must be raised to get that number. Therefore, using the property of logarithms, \[ \ln(e^x) = x \]we have:\[ \ln(e^{\sqrt{2}}) = \sqrt{2} \]

Step 2: Multiply by the Coefficient Outside the Logarithm

Now that we know \( \ln(e^{\sqrt{2}}) = \sqrt{2} \), we substitute it back into the original expression:\[ \sqrt{2} \ln(e^{\sqrt{2}) = \sqrt{2} \times \sqrt{2} \]Next, we need to multiply the terms on the right-hand side.

Step 3: Calculate the Final Result

Multiplying \( \sqrt{2} \) by itself results in:\[ \sqrt{2} \times \sqrt{2} = 2 \]This is because the square root of a number, when multiplied by itself, yields the original number. Thus the final result is 2.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a special logarithm with a base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. All logarithms involve understanding powers or exponents. The natural logarithm helps us solve equations involving exponential growth or decay.
For example, if you have \( \ln(e^x) \), it simplifies directly to \( x \). This is because the natural logarithm is the inverse function of exponentiation with base \( e \).
Let's break down why this is useful:

• The natural logarithm can unravel exponential expressions, giving insight into what the exponent is.
• This makes solving equations and evaluating expressions manageable without a calculator, as seen in \( \ln(e^{\sqrt{2}}) = \sqrt{2} \).

By understanding these fundamental ideas, we can greatly simplify otherwise complex expressions with natural logarithms.

Properties of Logarithms

Logarithms have specific properties that make them indispensable in simplifying mathematical expressions. When dealing with logarithms, especially natural logarithms, you can leverage several core properties:

• Power Rule: \( \ln(a^b) = b \cdot \ln(a) \). This lets us bring exponents down in front, simplifying the operations.
• Product Rule: \( \ln(xy) = \ln(x) + \ln(y) \). This property allows breaking down a product into a sum of logarithms, facilitating easier calculations.
• Quotient Rule: \( \ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y) \). This helps in simplifying division within logarithms.

In the given problem, we use the power rule to simplify \( \ln(e^{\sqrt{2}}) \) to \( \sqrt{2} \). This straightforward application exemplifies how knowing these properties can turn complicated expressions into simple ones, saving both time and effort.

Exponential Functions

An exponential function is of the form \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithm. Exponential functions grow rapidly and are used to model real-life situations such as population growth, radioactive decay, and interest calculations.
Understanding exponential functions involves recognizing how changes in the exponent affect the entire function. For instance, altering the exponent in \( e^x \) changes the rate of growth or decay exponentially.
Some key points about exponential functions include:

• Exponential Growth: If \( b > 0 \), the function represents exponential growth.
• Exponential Decay: If \( b < 0 \), the function depicts exponential decay.
• Inverse Relationship: Exponential functions and logarithms are inversely related. Hence, the natural logarithm undoes what an exponential function does.

This interplay between exponential functions and logarithms is crucial for solving various mathematical problems, including the simplification seen in the exercise, where the function \( e^{\sqrt{2}} \) is simplified using logarithmic properties.