Problem 53 - Precalculus with Limits

Question

Write the exponential equation in logarithmic form. \(e^{2}=7.3890 \ldots\)

Step-by-Step Solution

Answer

The logarithmic form of the equation \(e^{2}=7.3890 \ldots\) is \(\ln(7.3890) = 2\).

Step 1: Identify the Base, Exponent, and Result

In the given exponential equation, the base \(b\) is \(e\), the exponent \(y\) is 2, and the result or the output \(x\) is 7.3890.

Step 2: Convert to Logarithmic Form

The exponential equation \(b^{y} = x\) is equivalent to a logarithmic equation \(\log_{b}(x) = y\). Thus, by using the identified base, exponent, and result from Step 1, convert the exponential equation \(e^{2} = 7.3890\) into logarithmic form.

Step 3: Write the Logarithmic Form Equation

The logarithmic form of the equation is \(\log_{e}(7.3890) = 2\), which is conventionally written as \(\ln(7.3890) = 2\). Always remember that \(\log_{e}\) is also represented as \(\ln\), which is natural logarithm.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions are denoted as \( b^y \) where \( b \) is the base, and \( y \) is the exponent. A special and commonly used base for exponential functions is the mathematical constant \( e \), approximately equal to 2.71828. This base \( e \) is the foundation of natural exponential functions.

• In real-world scenarios, exponential functions often describe growth and decay processes, such as population growth, radioactive decay, and interest calculations.
• The general equation \( f(x) = a \cdot b^x \) defines an exponential function, where \( a \) is the initial amount, \( b \) is the growth factor, and \( x \) is time or any other variable indicating change.

Such functions are vital in understanding patterns that change at an increasing rate, making it essential for students to grasp their basic structure and applications.

Natural Logarithm

The natural logarithm, represented as \( \ln(x) \), is the logarithm to the base \( e \). It is the inverse function of the natural exponential function. When we say \( \ln(x) = y \), it implies that \( e^y = x \). This form of logarithm is widely utilized in calculus and higher mathematics due to the intrinsic properties of the base \( e \).

• Natural logarithms simplify the differentiation and integration of exponential functions, making them crucial for solving calculus problems involving growth and decay.
• The function \( \ln(x) \) is only defined for positive real numbers, reflecting the fact that you can't raise a positive number (\( e \)) to any power to get a negative or zero result.

Understanding natural logarithms is integral to transitioning from exponential expressions to their logarithmic counterparts within mathematical solutions.

Equation Conversion

Equation conversion from exponential form to logarithmic form helps solve equations involving exponentials, especially those that are tough to solve directly. In general, an exponential equation like \( b^y = x \) converts to the logarithmic form \( \log_b(x) = y \). This conversion allows us to isolate variables when they appear as exponents, a crucial step in solving many types of equations.

• To convert, identify the base, exponent, and result in the exponential equation.
• Use these to write the equivalent logarithmic form. For instance, \( e^2 = 7.3890 \) converts to \( \ln(7.3890) = 2 \).
• Conversion lets us comprehend and reason about problems that involve exponential growth and decay in a simpler format.

Converting between these forms enhances our mathematical toolkit, providing alternative perspectives and solutions to complex relationships within mathematics and its applications. Understanding this conversion is key to mastering equations encountered in a wide array of scientific and engineering fields.