Problem 39 - College Algebra

Question

For the following exercises, use the definition of common and natural logarithms to simplify. $$e^{\ln (1.06)}$$

Step-by-Step Solution

Answer

The expression simplifies to 1.06.

Step 1: Understanding the Exercise

The exercise asks us to simplify the expression \( e^{\ln(1.06)} \). We need to use the properties of logarithms and exponentials to do this.

Step 2: Identify the Property Used

Recognize that \( e^{\ln(x)} \) is a composition of an exponential function and a natural logarithm. One key property of logarithms and exponentials is that these functions are inverses of each other.

Step 3: Apply the Inverse Property

Since the exponential and the natural logarithm functions are inverses, we apply the property \( e^{\ln(x)} = x \). Therefore, in this case, \( e^{\ln(1.06)} = 1.06 \).

Step 4: Conclusion

After applying the inverse property of exponential and logarithmic functions, we conclude that the expression simplifies directly to 1.06.

Key Concepts

Exponential Functions

Exponential functions involve expressions where a constant base, often denoted as \( a \), is raised to a variable exponent, represented by \( x \). The general form of an exponential function is \( f(x) = a^x \). A crucial property of exponential functions is their rapid growth or decay:

• If the base \( a > 1 \), the function grows rapidly as \( x \) increases.
• If \( 0 < a < 1 \), the function decays rapidly as \( x \) increases.

The most commonly used exponential function in mathematics is when the base is Euler's number \( e \), approximately 2.718. This specific case is often written as \( f(x) = e^x \).
Exponential functions are vital in modeling real-life phenomena such as population growth, radioactive decay, and continuously compounded interest. Understanding the nature of exponential functions helps in making predictions and analyzing data in various scientific fields.

Inverse Property

In mathematics, the inverse property comes into play when dealing with operations that cancel each other out. In the context of exponential and logarithmic functions, this property is fundamental.
By definition, logarithms and exponentials are inverse operations. For example, the natural logarithm \( \ln(x) \) is the inverse of the exponential function \( e^x \). The inverse property tells us that applying one operation followed by its inverse returns us to the original value. This means:

• Applying the natural logarithm to an exponential results in the original exponent value, i.e., \( \ln(e^x) = x \).
• Similarly, raising \( e \) to the power of a natural logarithm brings us back to the original number, expressed as \( e^{\ln(x)} = x \).

This property is immensely useful in simplifying complex mathematical expressions, as seen in the example \( e^{\ln(1.06)} \), which simplifies directly to 1.06. Recognizing and applying the inverse property effectively bridges exponential and logarithmic transformations, aiding in solving equations and simplifying expressions.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. The logarithm of a number \( x \) with base \( a \) is written as \( \log_a(x) \), which answers the question: "To what power must \( a \) be raised, to produce \( x \)?"
The natural logarithm, denoted as \( \ln(x) \), is a logarithm with base \( e \).
Logarithmic functions are essential for solving exponential equations and transforming multiplicative processes into additive processes. This makes them perfect for simplifying calculations and modeling certain types of real-world problems.
Here are some important properties of logarithmic functions:

• The logarithm of the product is the sum of the logarithms: \( \log_a(xy) = \log_a(x) + \log_a(y) \)
• The logarithm of a quotient is the difference of the logarithms: \( \log_a\left( \frac{x}{y} \right) = \log_a(x) - \log_a(y) \)
• The logarithm of a power is the exponent times the logarithm: \( \log_a(x^b) = b \cdot \log_a(x) \)

Understanding these properties helps decode the nature of logarithmic functions and utilize them effectively for simplifying expressions like \( e^{\ln(1.06)} \). Recognizing how logarithms and exponentials interact, through properties like the inverse relationship, enhances problem-solving abilities in mathematics.