Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression. $$e^{\ln \left(x^{2}-3\right)}$$

Step-by-Step Solution

Answer

\(x^{2}-3\)

Step 1: Understand the expression

The expression \(e^{\ln \left(x^{2}-3\right)}\) is the exponential of the logarithm of \((x^{2}-3)\). The base of the exponential function is e (the Euler number, a mathematical constant approximately equal to 2.71828), and the logarithm is a natural logarithm (base e). In this expression, the exponential and the logarithm functions are inverse of each other.

Step 2: Apply the properties of logarithms

The properties of logarithms state that if you raise e to the power of a natural logarithm, you will get the argument of the logarithm itself. Therefore, \(e^{\ln \left(x^{2}-3\right)}\) simplifies to \(x^{2}-3\)

Key Concepts

Natural Logarithms

A natural logarithm is a logarithm with base 'e', where 'e' is an irrational constant approximately equal to 2.71828, known as Euler's number. Natural logarithms are denoted by the symbol \textbf{ln}. One of the fundamental relationships in mathematics involves the natural logarithm and the exponential function: for any positive number 'a', \( e^{\ln(a)} = a \) and conversely, \( \ln(e^a) = a \). This shows that the exponential function and the natural logarithm are inverse operations.

In practical problems, natural logarithms are used to solve equations where the variable is in the exponent of 'e', and to deal with continuous growth and decay models. The key to simplifying an expression like \( e^{\ln(x^{2}-3)} \) is recognizing that since 'e' is the base for both the exponential and the logarithm, applying the two operations in sequence effectively cancels them out, leaving just \( x^{2}-3 \).

Exponential Functions

Exponential functions are mathematical functions of the form \( f(x) = a^x \), where 'a' is a positive real number, and 'x' is any real number. The base 'a' determines the rate of growth or decay of the function when its variable 'x' changes. These functions are characterized by a constant rate of change percentage and are commonly used to model growth or decay in natural phenomena, finances, and other real-life situations.

The most important base for exponential functions in mathematics is the natural base 'e'. An exponential function with this base is written as \( e^x \) and has a rate of growth that's naturally occurring, often appearing in contexts such as continuously compounded interest or natural growth processes. The expression \( e^{\ln(x^{2}-3)} \) uses this natural base and exploits the unique trait of the exponential function: any quantity raised to the power of its natural logarithm results in the quantity itself.

Properties of Logarithms

The properties of logarithms are rules that simplify the process of manipulating logarithmic expressions. These properties are especially useful when you're trying to solve logarithmic equations or when working with exponential functions. Among the most crucial properties are:\

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• \Product Rule: \ln(a \times b) = \ln(a) + \ln(b)\
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• \Quotient Rule: \ln(\frac{a}{b}) = \ln(a) - \ln(b)\
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• \Power Rule: \ln(a^b) = b \times \ln(a)\
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• \Change of Base Rule: For bases a and b, \ln_b(a) = \frac{\ln(a)}{\ln(b)}\
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• \Inverse Property: \ln(e^a) = a and e^{\ln(a)} = a, for any a\
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In the original exercise, the inverse property is applied to the expression \( e^{\ln(x^2-3)} \) where 'e' and 'ln' cancel each other, resulting in \( x^2-3 \). Recognizing and applying these properties can help simplify complex logarithmic and exponential expressions and are critical tools for students tackling algebra and calculus problems.