Problem 54 - Precalculus with Limits

Question

Write the exponential equation in logarithmic form. \(e^{1 / 2}=1.6487 \ldots\)

Step-by-Step Solution

Answer

The logarithmic form of the provided exponential equation \( e^{1 / 2} = 1.6487 \ldots \) is \( \log_e 1.6487 \ldots = \frac{1}{2} \)

Step 1: Identifying Variables in the Given Exponential Form

Identify the base, exponent, and result in the given exponential form. In our question, \( e^{1 / 2} = 1.6487 \ldots \), 'e' is the base, \( \frac{1}{2} \) is the power (or exponent) and \( 1.6487 \ldots \) is the result.

Step 2: Converting to Logarithmic Form

Using the conversion rule, write the base as 'e' in log base, result as \( 1.6487 \ldots \) after the log base (or 'c' in our rule), and power as equal to the expression. This results in \( \log_e 1.6487 \ldots = \frac{1}{2} \)

Key Concepts

Exponential Equations

Exponential equations are mathematical statements that involve variables in the exponent position. They are written in the form \(b^x = y\), where \(b\) is the base, \(x\) is the exponent, and \(y\) is the result. In these equations, the base is raised to the power of the exponent to yield a particular result.
Understanding exponential equations is crucial as they appear frequently in various fields like science, engineering, and finance. They are used to model exponential growth or decay, such as population growth, radioactive decay, and interest calculations.
There are some important characteristics to remember about exponential equations:

• The base \(b\) must be a positive number, and often, it is a constant.
• The exponent \(x\) can be any real number—positive, negative, or even a fraction.
• As the exponent increases, the result generally grows rapidly, especially if the base is greater than one.

Converting these equations into another form, such as logarithmic form, can sometimes make solving them easier. This is especially useful when trying to find the value of the exponent.

Logarithms

A logarithm is the inverse operation to exponentiation. When we say \( \log_b(y) = x \), we mean that the base \(b\), raised to the power \(x\), equals \(y\). This is particularly useful for transforming complex exponential equations into a more manageable form.
In our given problem, we converted the exponential equation \(e^{1 / 2} = 1.6487\ldots\) into logarithmic form. This is how it works step by step:

• Identify the base: In our example, the base is \(e\), which is the natural base in mathematics.
• Convert the equation: We write \(\log_e(1.6487\ldots) = \frac{1}{2}\). Here, \(\log_e\) represents the natural logarithm (often written as \(\ln\)).

Using logarithms simplifies solving for exponents, especially when they are irrational or complicated. It tells us how many times we must multiply the base to obtain the given result.
Logarithms have several key properties that make them powerful:

• They convert multiplication into addition, which is handy for calculations.
• They provide a way to solve equations where the unknown is an exponent.
• They allow comparison of very large or very small numbers more easily.

Mathematical Notation

Mathematical notation is a system of symbols used to represent numbers, variables, operations, and relationships in mathematics. It provides a universal language for mathematicians to communicate ideas efficiently and precisely.
For this exercise, understanding the notation behind exponential equations and logarithms is critical.

• Exponential Notation: This involves writing numbers as a base raised to an exponent, such as \(b^x = y\). This denotes that the number \(b\) is multiplied by itself \(x\) times to give \(y\).
• Logarithmic Notation: In this form, \(\log_b(y) = x\) indicates that the base \(b\) has been raised to the power of \(x\) to get \(y\). This is particularly useful for reversing an exponentiation operation.
• Ellipsis (\(\ldots\)): Used in numbers like \(1.6487\ldots\), indicating that the number is non-terminating or is to be understood as a continuation.

Clear mathematical notation is essential for accuracy in solving problems and communicating solutions. It minimizes misunderstandings and clarifies each step in solving mathematical exercises.