Problem 94 - College Algebra

Question

Evaluate or simplify each expression without using a calculator. $$ e^{\ln 300} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the expression

The expression given is \( e^{\ln 300} \). This involves both exponential function and natural logarithmic function. According to the property of natural logarithm, the base 'e' and the exponent 'Ln 300', in this expression will cancel out each other when they are used in this specific combination as an exponentiation

Step 2: Applying the property of natural logarithms

Using the property of natural logarithms, \( e^{\ln a} = a \), where a is any number, the given expression simplifies to 300

Key Concepts

Exponential Functions

Exponential functions are an essential piece of the mathematical puzzle, particularly when it comes to growth patterns and decay in various fields like economics, biology, and physics. An exponential function can be expressed in the form of
\( f(x) = a \times b^x \)
where 'a' is a constant, 'b' is the base and 'x' is the exponent. A special exponential function is
\( e^x \)
, where 'e' is Euler's number, approximately equal to 2.71828. It's known for its unique property of being the rate of growth shared by all continually growing processes. When you work with
\( e^x \)
, it represents continuous growth at a rate of 100% per unit time period, a model that's crucial in various mathematical and real-life applications, such as compound interest or natural growth processes.

It is vital to understand the relationship between exponential functions and logarithms. In the homework exercise,
\( e^{\text{ln} 300} \)
represents such a relationship. This equation makes use of the natural logarithm of 300, and it ties back to one of the most fundamental identities concerning exponential and logarithmic functions. The exercise solution reveals the property that logarithms are the inverses of exponentials, thus they 'cancel' each other out in specific situations.

Logarithmic Properties

The properties of logarithms are the rules that allow us to manipulate logarithmic expressions efficiently and follow some simple patterns. The key property used in the original exercise is the inverse relationship between logarithms and exponentials.

For any positive real number 'a', and 'b' > 0, 'b' ≠ 1, the logarithm is the power to which 'b' must be raised to obtain 'a'. It is denoted by
\( \text{log}_b(a) \)
. The natural logarithm \( \text{ln}(x) = \text{log}_e(x) \)
uses 'e' as its base and is widely used due to its natural properties in calculus and exponential growth models.

One of the fundamental logarithmic properties that apply to natural logarithms is that
\( e^{\text{ln}(x)} = x \)
and
\( \text{ln}(e^x) = x \)
, which shows how they serve as opposites to each other, allowing simplification of expressions that contain both an 'e' and a natural log. In the exercise, this property is used to simplify
\( e^{\text{ln} 300} \)
directly to 300, showcasing a clear example of how this property operates.

Simplifying Expressions

Simplifying expressions is a vital skill in mathematics, turning complex-looking equations into more manageable forms. This simplification often involves applying various arithmetic rules and properties of algebraic operations. In the original exercise, the simplification process centers on understanding the relationship between an exponent and a logarithm.

When simplifying expressions, especially those including exponents and logarithms, one must identify and apply the correct properties. In the exercise, recognizing the inverse relationship between the base 'e' of the exponential function and the natural logarithm allows for immediate simplification. Here, instead of performing complex operations, recognizing the pattern leads to a swift and elegant solution - arriving at the simple number 300 from the seemingly complex expression
\( e^{\text{ln} 300} \)
.

Approaching mathematics with the goal of simplification can often lead to deeper understanding and elegance in solutions. By mastering these concepts, one can move through problems more efficiently, laying the groundwork for tackling more advanced mathematics with ease.