Problem 86 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of natural logarithms to rewrite the expression. $$e^{\ln 22}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(e^{\ln 22}\) is 22.

Step 1: Identify the expression

The expression given in the problem is \(e^{\ln 22}\).

Step 2: Apply the property of natural logarithms

Apply the property \(e^{\ln a} = a\) to the given expression. Here, a is 22.

Step 3: Simplify the expression

By replacing a with 22 in the property, we get \( e^{\ln 22} = 22\). So, the given expression simplifies to 22.

Key Concepts

Natural Logarithm Simplification

Understanding how to simplify natural logarithms is crucial for students who are working with exponential and logarithmic expressions. The natural logarithm, represented by \(\ln x\), is the power to which the base \(e\), Euler's number (approximately 2.71828), must be raised to obtain the number \(x\).

Simplification of a natural logarithm often involves applying fundamental properties and recognizing specific patterns allows for easy simplification. For instance, one essential property is \(e^{\ln x} = x\), which arises from the inverse relationship between \(e^x\) and \(\ln x\). In the exercise example \(e^{\ln 22}\), by applying this property, we can see that the expression simplifies directly to 22 without requiring complex calculations.

This property is particularly useful because it can simplify expressions where the base \(e\) of an exponential function is raised to a logarithmic function with the same base. The understanding of this simplification process not only makes it easier to work with natural logarithms but also reinforces the concept that logarithmic and exponential functions are inverse operations.

Exponential Functions

Exponential functions play a vital role in various fields including science, finance, and engineering. They are defined by the formula \(f(x) = a \cdot e^{bx}\), where \(e\) is the base of the natural logarithm, and \(a\) and \(b\) are constants. The variable \(x\) often represents time, growth, or decay depending on the context.

What characterizes an exponential function is its rapidly increasing or decreasing nature, and how it models phenomena such as population growth, radioactive decay, and continuously compounded interest.

An important aspect of working with exponential functions is understanding their properties, such as the fact that the function \((e^x)\) always yields a positive value for any real number \(x\). This means exponential functions never cross the x-axis when graphed and they have a horizontal asymptote at \(y = 0\). Recognizing these characteristics is vital in solving and graphing exponential functions.

Logarithmic and Exponential Properties

Logarithmic and exponential functions are deeply interconnected, as logarithms are the inverses of exponentials. This inverse relationship is highlighted through their properties which are foundational in simplifying complex expressions.

Some key logarithmic properties include:

• \(\ln(e) = 1\): Recognizing \(e\) as the base of the natural logarithm.
• \(\ln(1) = 0\): The log of one is always zero, regardless of the logarithm's base.
• Product Rule: \(\ln(xy) = \ln(x) + \ln(y)\)
• Quotient Rule: \(\ln(\frac{x}{y}) = \ln(x) - \ln(y)\)
• Power Rule: \(\ln(x^p) = p \cdot \ln(x)\)

Exponential properties include:

• Any number raised to the power of zero is one: \(b^0 = 1\).
• For any positive number \(a\), \(a^1 = a\).
• The rate of change of an exponential function, given by \(\frac{d}{dx}(e^x)\), is \((e^x)\), illustrating the function's property of continuous growth.

By mastering these properties, students can tackle a broad range of problems involving natural logarithms and exponential functions, including growth and decay models, logarithmic differentiation, and solving equations analytically. This fundamental knowledge fosters a deeper comprehension of how these mathematical tools describe real-world scenarios.