Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$\ln e^{\sqrt{6}}$$

Step-by-Step Solution

Answer

\( \sqrt{6} \)

Step 1: Understanding the Natural Logarithm

The natural logarithm function, denoted as \( \ln \), is the inverse of the exponential function \( e^x \). Thus, \( \ln(e^x) = x \) for any real number \( x \). In this expression, we have \( x = \sqrt{6} \).

Step 2: Applying the Inverse Property

Given the expression \( \ln(e^{\sqrt{6}}) \), due to the inverse property of logarithms and exponentials, you can simplify this to just \( \sqrt{6} \). This is because applying the natural logarithm to \( e \) raised to any power will simply return that power.

Key Concepts

inverse functions

Inverse functions are a pair of functions that essentially "undo" each other. For example, if you have a function \( f(x) \) and its inverse \( f^{-1}(x) \), applying \( f \) and then \( f^{-1} \) brings you back to your original value. Mathematically, this is expressed as:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

In the context of natural logarithms and exponential functions, the exponential function \( e^x \) and the natural logarithm \( \ln(x) \) are inverses of each other. This means when you take the natural logarithm of \( e \) raised to a power, you simply get back the power itself:

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

This inverse relationship is what allows us to simplify expressions like \( \ln(e^{\sqrt{6}}) \) to \( \sqrt{6} \). Understanding inverse functions is crucial when working with logarithms and exponents, as it allows you to quickly simplify complex expressions.

exponential function

The exponential function plays a significant role in both mathematics and real-world applications. It is denoted as \( e^x \), where \( e \) is a constant approximately equal to 2.71828. This function grows rapidly as the exponent \( x \) increases, making it essential for modeling phenomena involving growth or decay, such as populations or radioactive decay. A key property of the exponential function is its behavior under the inverse operation provided by natural logarithms. Since the exponential function is the inverse of the natural logarithm, applying both in succession will return the original value. This aligns with the property \( \ln(e^x) = x \), which is used to simplify complex algebraic expressions quickly. Another special feature is that the derivative of \( e^x \) is itself \( e^x \), a unique property that makes it extremely important in calculus. Thus, understanding the exponential function and its inverse relationships is fundamental for simplifying and solving mathematical problems involving growth, decay, and other exponential processes.

properties of logarithms

Understanding the properties of logarithms is critical for simplifying and solving equations involving logarithmic terms. Logarithms transform multiplicative relationships into additive ones, which can simplify calculations significantly.Here are some important properties:

• Product Rule: \( \ln(AB) = \ln(A) + \ln(B) \)
• Quotient Rule: \( \ln\left(\frac{A}{B}\right) = \ln(A) - \ln(B) \)
• Power Rule: \( \ln(A^B) = B\ln(A) \)

These rules apply not only to natural logarithms but to logarithms in any base. However, the natural logarithm has a unique inverse relationship with the exponential function, given by \( \ln(e^x) = x \), which can be applied directly in many scenarios, like simplifying \( \ln(e^{\sqrt{6}}) \) to \( \sqrt{6} \).By understanding these properties, you can tackle complex expressions containing logs and exponentials with more confidence and ease.