Problem 26 - Algebra and Trigonometry

Question

Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } e^{\ln \pi}} & {\text { (b) } 10^{\log 5}} & {\text { (c) } 10^{\log .87}}\end{array} $$

Step-by-Step Solution

Answer

(a) \(\pi\), (b) 5, (c) 0.87

Step 1: Understanding the Expression

The expression involves evaluating the natural exponential function and logarithmic operations. The task is to simplify each part given in forms like \( e^{\ln x} \) and \( 10^{\log x} \).

Step 2: Simplifying Part (a)

For \( e^{\ln \pi} \), recall the property \( e^{\ln x} = x \). By this property, we can simplify \( e^{\ln \pi} \) to \( \pi \).

Step 3: Simplifying Part (b)

For \( 10^{\log 5} \), we use the property \( 10^{\log x} = x \). Thus, \( 10^{\log 5} \) simplifies to \( 5 \).

Step 4: Simplifying Part (c)

For \( 10^{\log 0.87} \), apply the same property \( 10^{\log x} = x \). Thus \( 10^{\log 0.87} \) simplifies to \( 0.87 \).

Key Concepts

Natural Exponential Function

The natural exponential function is a fascinating and widely used mathematical concept. It involves the number "e" (approximately 2.71828), which is an irrational constant and serves as the base for the natural logarithms. When we express something like \( e^{\ln x} \), the beauty of this expression lies in its elegant simplicity.

• The "e" in the expression is the base, while the "\( \ln x \)" is the power or exponent.
• \( \ln x \) signifies the natural logarithm of "x", which is a special logarithm with base "e".

A key property to remember is that \( e^{\ln x} = x \). This property implies that when you raise "e" to the power of the natural logarithm of "x", the result is simply "x". This makes the simplification of expressions like \( e^{\ln \pi} \) straightforward. Using the property, \( e^{\ln \pi} \) directly simplifies to \( \pi \). This concept is essential in calculus, differential equations, and other fields requiring logarithmic and exponential computations.

Logarithmic Operations

Understanding logarithmic operations is critical to simplifying complex expressions. Logs are inverse operations to exponentiation and provide a way to deal with very large or very small numbers efficiently. Logarithms have different bases, but when it comes to the base 10 logarithm (common logarithm), represented as \( \log x \), some straightforward properties can be applied.

• The expression \( 10^{\log x} \) uses base 10 for both the logarithm and the exponent.
• By the property \( 10^{\log x} = x \), whenever the operands have matching bases like 10, the operation simplifies neatly to "x".

For example, to solve \( 10^{\log 5} \), you directly get 5, and similarly for \( 10^{\log 0.87} \), the result is 0.87. Recognizing this property allows us to evaluate and simplify these expressions quickly without extensive calculations, making logarithmic operations powerful in mathematical analysis.

Simplification of Expressions

Simplifying expressions involves using mathematical properties and operations to transform them into their simplest forms. It minimizes complexity and helps in understanding the underlying structure of the expressions. This is especially useful when dealing with exponential and logarithmic expressions.

• Simplification often involves applying specific properties or rules, such as \( e^{\ln x} = x \) or \( 10^{\log x} = x \), to reduce the expression to a more manageable form.
• It can eliminate unnecessary complexity, making it easier to evaluate or compare expressions.

In our exercise, we simplified \( e^{\ln \pi} \) to \( \pi \), \( 10^{\log 5} \) to 5, and \( 10^{\log 0.87} \) to 0.87 by leveraging these properties. Effective simplification reduces error margins in calculations and enhances conceptual clarity, fostering a deeper comprehension of the subject.