Problem 18 - Algebra and Trigonometry

Question

\(7-18\) Evaluate the expression. $$ \ln \left(\ln e^{e^{200}}\right) $$

Step-by-Step Solution

Answer

The expression evaluates to 200.

Step 1: Understand the Expression

The expression we need to evaluate is \( \ln(\ln e^{e^{200}}) \). This means we first need to find \( \ln e^{e^{200}} \), and then take the natural logarithm of that result.

Step 2: Simplify the Inner Exponent

Recall that the natural logarithm \( \ln a = b \) implies that \( e^b = a \). Thus, for the innermost part \( e^{e^{200}} \), we note that if we take the natural logarithm of \( e^{e^{200}} \), we simply retrieve the exponent: \( \ln(e^{e^{200}}) = e^{200} \).

Step 3: Evaluate the Outer Logarithm

Now, we need to find \( \ln(e^{200}) \). Similarly, by the property of logarithms, the natural logarithm of \( e \) raised to any power returns the exponent itself. Therefore, \( \ln(e^{200}) = 200 \).

Key Concepts

Exponential Function

The concept of an exponential function is foundational to understanding many mathematical expressions, particularly those involving growth and decay. An exponential function can be generally represented as \( f(x) = a^x \), where \( a \) is a constant and \( x \) is a variable. This constant \( a \) is known as the base of the exponential. The base is often the number \( e \), which is approximately equal to 2.718. The function \( e^x \) is referred to as the natural exponential function.

Exponential functions possess unique characteristics, such as:

• They are always positive for any real number \( x \).
• Their rate of growth or decay is proportional to the value of the function.
• Their graphs smoothly increase (or decrease) without any breaks.

Understanding these properties can help in visualizing how expressions such as \( e^{e^{200}} \) behave. Here, "power of an exponent" is a key concept, whereby the exponent itself is an exponential expression. This is a higher level of exponential growth.

Recognizing how to manipulate and simplify exponential expressions by employing natural logarithms is critical in many advanced mathematical and real-world applications.

Logarithmic Properties

Logarithms, especially natural logarithms denoted as \( \ln \), have specific properties that make them powerful tools in algebra. The natural logarithm of a number \( a \) is the power you raise \( e \) to, in order to get \( a \).

Several key properties of logarithms can help simplify complex expressions:

• Product Property: \( \ln(xy) = \ln x + \ln y \)
• Quotient Property: \( \ln\left(\frac{x}{y}\right) = \ln x - \ln y \)
• Power Property: \( \ln(x^a) = a \ln x \)

In the context of the original exercise, knowing that \( \ln(e^a) = a \) directly applies in both steps. First, it simplifies \( \ln(e^{e^{200}}) = e^{200} \) and then \( \ln(e^{200}) = 200 \).

Understanding and applying these properties provides an efficient way to simplify otherwise dauntingly large numbers or expressions, allowing for easier manipulation and clear solution paths.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations, representing a mathematical relationship. They serve as the basis for formulating equations and allowing for solving various types of problems in mathematics.

In the context of logarithmic and exponential problems, managing algebraic expressions involves applying rules of arithmetic and algebra, including:

• Combining like terms—terms with the same base and exponent.
• Distributing multiplication over addition or subtraction when necessary—using techniques such as the distributive property.
• Finding common factors and simplifying expressions when applicable.

Each algebraic expression can be simplified through strategic manipulation. For instance, the expression \( \ln(e^{e^{200}}) \) is simplified by recognizing that the natural logarithm cancels out the base \( e \), peering directly at the exponents.

By leveraging algebraic skills, you can tweak even the most complex expressions into comprehensible formats, ultimately easing the problem-solving process and strengthening mathematical intuition.