Problem 97 - Algebra and Trigonometry

Question

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ e^{\ln 5 x^{2}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(e^{\ln 5 x^{2}}\) is \(5 x^{2}\)

Step 1: Understand the problem and properties

Required is to simplify the expression \(e^{\ln 5 x^{2}}\). It is crucial to remember the property that \(e^{\ln a} = a\). Here \(a\) represents \(5 x^{2}\).

Step 2: Apply the property

Substitute \(a\) with \(5 x^{2}\) in the property \(e^{\ln a} = a\). Therefore the expression \(e^{\ln 5 x^{2}}\) simplifies to \(5 x^{2}\)

Key Concepts

Simplification

The term "simplification" in mathematics refers to the process of reducing an expression to its simplest form. This involves combining like terms, factoring, and applying various mathematical properties to make an expression more understandable and easier to work with. When it comes to exponential and logarithmic expressions, simplification often requires the use of specific properties and rules. The expression given in this exercise, \(e^{\ln 5 x^{2}}\), can be daunting if you aren't familiar with these rules. Simplifying an expression means to eliminate any complex terms, like logarithms or exponents, that can be reduced to simpler forms. By understanding how properties such as \(e^{\ln a} = a\) work, we can quickly "cancel out" the log and the base \(e\) in this expression, simplifying it directly down to \(5x^{2}\). This process not only reduces the complexity of the expression but also helps in understanding the role of these terms in the context of a broader problem.

Natural Logarithm

The natural logarithm, denoted as \(\ln x\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. This type of logarithm is particularly important in continuous growth processes and is used extensively in calculus and exponential growth equations.Understanding the natural logarithm involves knowing that it is the inverse operation of exponentiation with base \(e\). For example, the expression \(e^{\ln a}\) simplifies to \(a\). This important property allows us to manipulate and simplify expressions involving exponential and logarithmic terms efficiently.When simplifying expressions like \(e^{\ln 5 x^{2}}\), it is crucial to recognize that the \(\ln\) and \(e\) functions cancel each other out. This is because the natural logarithm measures the amount of time needed for growth to reach a certain level, and \(e\) represents the compounding factor needed to achieve that growth, thus bringing us back to the original quantity inside the logarithm, \(5x^2\). This property is a cornerstone of many mathematical simplifications and calculations.

Properties of Exponents

The properties of exponents play a vital role in simplifying mathematical expressions. Exponents indicate repeated multiplication of a base number, and understanding their properties helps in unraveling complex expressions.Some key properties include:

• \(a^{m} \cdot a^{n} = a^{m+n}\): This property allows you to add exponents when multiplying like bases.
• \((a^{m})^{n} = a^{mn}\): This property occurs when you raise a power to another power, multiplying the exponents together.
• \(a^{-n} = \frac{1}{a^n}\): This property is used to rewrite negative exponents by taking the reciprocal of the base raised to the positive exponent.

In the context of logarithms, especially, one very useful property is \(e^{\ln a} = a\). This comes from recognizing that the operations \(e\) and \(\ln\) are inverses, and thus cancel each other out, simplifying the expression to just the base \(a\).By mastering these properties, it is easier to approach problems involving exponents and logarithms, like in the original problem, and manipulate them more effectively to achieve the simplest form possible.