Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$e^{3 / 4}=2.1170. . . $$

Step-by-Step Solution

Answer

The logarithmic form of the equation \(e^{3 / 4}=2.1170 . . . \) is \(\ln (2.1170 . . . )=3/4\).

Step 1: Identify the base, exponent and result of the exponential form

The exponential equation is given as \(e^{3 / 4}=2.1170 . . . \). Here \(e\) is the base, \(3/4\) is the exponent, and \(2.1170 . . . \) is the result. Our goal is to express this relationship in logarithmic form.

Step 2: Apply the logarithmic form rule

The equivalent logarithmic representation of an exponential equation with form \(b^{y}=x\) is \(\log_{b}x=y\). In our case, \(b=e\), \(y=3/4\), and \(x=2.1170 . . . \). By replacing these into our logarithmic equation, we end up with \(\log_{e} (2.1170 . . . )=3/4\).

Step 3: Write the final logarithmic form

Specifically for the base \(e\), the logarithm is written as ln. So the final logarithmic form of the given equation would be \(\ln (2.1170 . . . )=3/4\).

Key Concepts

Understanding Exponential Equations

An exponential equation is a type of mathematical equation where a constant base is raised to a variable exponent. These equations have the general form \( b^y = x \), where:

• \( b \) is the base, typically a constant like 2, 10, or \( e \), the natural number.
• \( y \) is the exponent or power to which the base is raised.
• \( x \) represents the result or the outcome of the exponential operation.

The exponential equation is a powerful tool in mathematics because it can describe natural processes like population growth, radioactive decay, and interest calculations.

Exploring Base and Exponent

In the realm of exponential equations, the terms "base" and "exponent" are crucial.The base is the number that gets multiplied by itself. In expressions like \( e^{3/4} \), \( e \) is the base, the mathematical constant approximately equal to 2.71828.
The exponent indicates how many times the base is used as a factor. For example, in \( e^{3/4} \), 3/4 is the exponent. It tells you that \( e \) is multiplied "root-ishly" 3/4 times. When converting exponential forms to logarithmic forms, these components play a significant role in reshaping the expression.

The Role of the Natural Logarithm

The natural logarithm, commonly denoted as \( \ln \), is a logarithm to the base\( e \). It simplifies expressions and equations involving the constant \( e \).The natural logarithm is a handy tool in calculus and different branches of science.
In the context of the given equation \( e^{3/4} = 2.1170 \ldots \), we use \( \ln \) to rewrite the equation in logarithmic form, resulting in \( \ln(2.1170 \ldots) = 3/4 \). This expression tells us the power to which we elevate \( e \) to achieve the number 2.1170.

Conversion Processes in Math

When dealing with equations, understanding conversion processes is essential. In this context, conversion is used to switch between exponential and logarithmic forms.

• To convert an exponential equation \( b^y = x \) to its logarithmic equivalent, use the form \( \log_b(x) = y \).
• Here, \( b \) is the base in both forms, \( y \) is the exponent in the exponential form, and the result in the logarithmic form.
• In our example, \( e^{3/4} = 2.1170 \ldots \) converts to \( \ln(2.1170 \ldots) = 3/4 \).

This conversion process allows us to work with difficult-to-handle exponential equations more easily.