Question

In Exercises, apply the inverse properties of logarithmic and exponential functions to simplify the expression. $$ e^{\ln (5 x+2)} $$

Short answer

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

Step-by-step solution

Identify inverse functions

First, identify the inverse functions in the given expression: \( e \) and \( \ln \). They occur together \( e^{\ln (5 x+2)} \) in the given expression.

Apply inverse operation

The next step is to apply the property that exponential function and logarithm are inverse processes. Therefore, the expression \( e^{\ln (5 x+2)} \) simplifies to \( 5x+2 \).

Key concepts

Logarithmic Functions

Logarithmic functions play a crucial role in mathematics, especially in solving problems involving exponential growth or decay. These functions are the inverse of exponential functions. For example, the natural logarithm, denoted as \( \ln \), is the inverse of the exponential function \( e^x \). When you see an expression with a logarithm, like \( \ln(y) \), it asks the question: "What power should \( e \) be raised to, in order to get \( y \)?"

• Logarithms convert multiplication into addition, which simplifies complex computations.
• The natural logarithm \( \ln \) is particularly important due to its base of \( e \), approximately 2.71828.
• Logarithmic properties such as \( \ln(ab) = \ln(a) + \ln(b) \) and \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \) help in simplifying expressions.

In the context of the exercise, recognizing that \( \ln \) and \( e \) are inverse functions is the key to simplification.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent. The form \( e^x \) represents a special exponential function where the base is the mathematical constant \( e \), approximately 2.71828.

• The natural exponential function, \( e^x \), grows at a rate proportional to its value, making it important in modeling continuous growth processes.
• Exponential growth can describe populations, investments, and more, leading to rapid increases over time.
• Inverse functions, like \( \ln \), provide a means to "undo" exponential functions, simplifying expressions.

In the given exercise, the expression \( e^{\ln(5x + 2)} \) shows how applying the exponential function \( e \) and its inverse, the logarithm \( \ln \), directly undoes each other, leading to simplification.

Simplifying Expressions

When simplifying expressions, identifying and employing inverse properties is fundamental. Especially in expressions involving exponential and logarithmic components, understanding how these functions interact paves the way for simplification.

• The primary goal is to reduce complex expressions to simpler forms without changing their value.
• Recognize inverse pairs, such as \( e^{\ln(x)} = x \), to directly simplify expressions.
• Use recognition: expressions involving both a logarithm and an exponential of the same base can typically be simplified by canceling these functions.

In this context, the expression \( e^{\ln(5x + 2)} \) leverages the inverse property: the exponential function \( e \) "cancels out" the natural logarithm \( \ln \). This results in the straightforward simplified expression \( 5x + 2 \).