Problem 12 - Algebra and Trigonometry

Question

Evaluate the expression. $$ \ln \left(\ln e^{e^{-200}}\right) $$

Step-by-Step Solution

Answer

The expression evaluates to -200.

Step 1: Evaluate the Inner Exponent

Start by evaluating the expression inside the innermost parentheses. The innermost expression is \( e^{-200} \). There's nothing to simplify here as it's already in its simplest 'e' power form.

Step 2: Simplification Using Logarithm Properties

Apply the property of logarithms, \( \ln(e^x) = x \), to simplify the nested logarithm. So, \( \ln(e^{e^{-200}}) \) simplifies to \( e^{-200} \). This is because the logarithm and the exponent are inverse functions.

Step 3: Evaluate the Outer Logarithm

Now, evaluate the outer logarithm. We need to find \( \ln(e^{-200}) \). Using the same logarithm property, \( \ln(e^x) = x \), this expression simplifies further to \(-200\).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a mathematical function with a special base: the constant \( e \), approximately equal to 2.71828. Unlike other logarithms, which may have arbitrary bases, the natural logarithm focuses on \( e \) due to its natural occurrence in various scientific and mathematical contexts. It's heavily used in exponential growth and decay problems, as well as in calculus.
Natural logarithms have the property of being the inverse function to the exponential function \( e^x \). This means that if you have an expression such as \( \ln(e^x) \), it simplifies directly to just \( x \). This inverse nature is a central concept in working with natural logarithms.
When evaluating expressions like \( \ln(e^{e^{-200}}) \), the natural logarithm allows us to simplify the term "inside" the logarithmic function quickly and efficiently, as seen in the solution with steps.

Logarithm Properties

Logarithms have several fundamental properties which can greatly simplify expressions and calculations. Let's explore these properties and see how they help solve problems involving logarithms, such as the original exercise.

• The inverse property: \( \ln(e^x) = x \). This tells us that the natural logarithm undoes the action of raising \( e \) to a power. Hence, it can quickly simplify expressions where \( e \) is raised to any power.
• The product property: \( \ln(xy) = \ln(x) + \ln(y) \). This allows you to split the logarithm of a product into a sum of logarithms.
• The quotient property: \( \ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y) \). This helps in breaking down complex divisions into simpler subtractions of logarithms.
• The power property: \( \ln(x^y) = y\ln(x) \). This lets you bring a power inside the natural logarithm out front, greatly simplifying multiplications into scalable additions.

By leveraging these properties, particularly the inverse property, evaluating expressions like \( \ln(e^{-200}) \) becomes straightforward, as it simplifies directly to \(-200\).

Exponential Functions

Exponential functions express the concept of constant percentage change and growth in mathematical form, commonly noted as \( e^x \). These functions are crucial in numerous real-world applications—such as calculating compound interest, population growth, and radioactive decay—because of their versatile nature.
In the context of logarithmic expressions, exponential functions are the reverse of logarithms. They act as the "undo" button for logarithmic equations, as seen with the key property \( \ln(e^x) = x \). This inverse relationship means that applying logarithms to an exponential function will effectively bring down the exponent as an answer.
Understanding this connection provides great efficiency when working with nested expressions like \( e^{e^{-200}} \). By applying the logarithm properties and simplifying, we discover answers that reveal the inherent behavior and transformation of exponential growth or decay more clearly.