Problem 94 - Algebra and Trigonometry

Question

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ e^{\ln 300} $$

Step-by-Step Solution

Answer

The simplified form of the expression \( e^{\ln 300} \) is 300.

Step 1: Recognize the property of logarithms

To begin with, recognize the property of logarithms which states that \( e^{\ln b} = b \) for any positive number b, given that \( e \) is the base of the natural logarithm. This is because the exponential function and the logarithm function are inverse of each other. As a result, they cancel each other when applied consecutively.

Step 2: Apply the property of logarithms

By recognizing this property, the given expression \( e^{\ln 300} \) can now be simplified by replacing b in the recognized property with 300 to yield 300.

Key Concepts

Exponential Functions

Exponential functions are vital in mathematics and represent a wide array of real-world phenomena. An exponential function can be expressed in the form of \( f(x) = a\cdot e^{bx} \), where \( a \) and \( b \) are constants and \( e \) is a mathematical constant approximately equal to 2.71828. The exponent \( bx \) is a variable, making the function grow or decay based on its value.

Here are some key points to remember about exponential functions:

• Exponential Growth: This occurs when the growth factor \( b \) is positive, leading to a rapidly increasing function.
• Exponential Decay: In contrast, this happens when \( b \) is negative, resulting in a decreasing function.
• The Natural Base: The number \( e \) is known as the "natural base," and exponential functions with \( e \) are termed "natural exponential functions." These are commonly encountered in calculus and higher mathematics.

Understanding exponential functions helps in analyzing patterns in finance, biology, and many other fields.

Inverse Functions

Inverse functions are crucial in understanding how different mathematical operations relate to each other. An inverse function essentially reverses the operation of another function. If \( f(x) \) is a function, then its inverse \( f^{-1}(x) \) satisfies the condition \( f(f^{-1}(x)) = x \), for all \( x \) in the domain of \( f^{-1} \).

This relationship is prominent in the context of exponential and logarithmic functions.

• The Exponential and Logarithmic Relationship: The logarithm is an inverse operation to exponentiation. This means \( e^{\ln b} = b \), because the exponential function \( e^x \) and the natural logarithm \( \ln x \) negate each other's effect.
• Graphical Representation: The graph of a function and its inverse is symmetric over the line \( y = x \).
• Finding Inverses: To find an inverse function, swap the roles of \( x \) and \( y \) in the equation and solve for \( y \).

The concept of inverse functions allows us to solve equations and understand the interconnectedness of different mathematical operations.

Simplifying Expressions

Simplifying expressions involves reducing complex mathematical statements into simpler forms to make calculations easier or to identify specific properties. This exercise often requires applying several mathematical properties or identities.

When simplifying the expression \( e^{\ln 300} \), you employ properties to make the simplification straightforward:

• Utilizing Inverse Properties: By recognizing that \( e \) and \( \ln \) are inverse operations, you can directly simplify \( e^{\ln 300} \) to \( 300 \).
• Steps to Simplify: Identify complementary functions (like \( e^x \) and \( \ln x \)) that cancel each other, leading you to simpler expressions.
• Simplification in Practice: This skill is useful in algebra and calculus to solve equations efficiently and understand deeper mathematical insights.

Simplifying expressions can save time and provide clearer understanding in manipulating and interpreting mathematical statements.