Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln \frac{1}{e^{2}}=-2$$

Step-by-Step Solution

Answer

The exponential form of the equation \(\ln \frac{1}{e^{2}}=-2\) is \( e^{-2} = \frac{1}{e^{2}} \).

Step 1: Understand the given logarithmic equation

The provided equation is \( \ln \frac{1}{e^{2}}=-2 \). This equation is in logarithmic form using the natural log function denoted by \(\ln\), which is equivalent to the logarithm with base \(e\), where \(e\) is the Euler's number, approximately equal to 2.718.

Step 2: Apply logarithmic rule to transform into exponential form

The logarithm rule states that for any \(x\) and \(b\), \(b^{\log_bx}=x\). A logarithmic equation \( \log_b a = c \) can be rewritten in exponential form as \( b^{c} = a \). Applying this idea, the equation becomes: \( e^{-2} = \frac{1}{e^{2}} \). This is the equivalent expression in exponential form.

Key Concepts

Natural Logarithm

The natural logarithm is a special type of logarithm where the base is the mathematical constant \(e\). This constant is approximately equal to 2.718 and is known as Euler's number. Natural logarithms are used widely in science, engineering, and mathematics since they have many unique properties that simplify equations and calculations. Whenever you see \( \ln\), it indicates the use of the natural logarithm. So, when you have an equation such as \( \ln x = y\), it is similar to saying the power to which \(e\) must be raised to get \(x\) equals \(y\). This foundational idea allows us to convert between exponential and logarithmic forms easily.

In the given exercise, \( \ln \frac{1}{e^2} = -2\) means we are looking for the exponent that gives us \(\frac{1}{e^2}\) when \(e\) is raised to that power. Understanding that \(\ln\) stands for logarithm with base \(e\) makes it possible to interpret and solve such equations quickly.

Exponential Equations

Exponential equations involve functions where the variable appears in the exponent. These equations often involve the mathematical constant \(e\), particularly when you're dealing with natural logarithms. Converting logarithmic expressions to exponential form involves identifying the base, which in our example is \(e\), and rewriting the equation to show how some number is equivalent to the base raised to a power.

For example, the given equation \( \ln \frac{1}{e^2}=-2\) can be rewritten as an exponential equation: \(e^{-2} = \frac{1}{e^2}\). This confirms that the equation is true, as \(e\) raised to the power of \(-2\) indeed equals \(\frac{1}{e^2}\). By converting from logarithmic to exponential form, we simplify the structure and can further understand the relationship between the numbers involved.

• An exponential form can help in visualizing the growth or decay process in real-life applications, such as exponential growth in populations or radioactive decay.

Logarithmic Rules

Logarithmic rules are essential tools for manipulating and understanding logarithmic and exponential expressions. One of the fundamental properties is understanding that logarithms and exponents are inverse processes. This means any equation in logarithmic form can be converted into exponential form and vice versa, thanks to this inverse relationship.

One of the primary rules used in the given solution is the change of form from logarithmic to exponential. The rule, \( \log_b a = c \) can be translated to \( b^c = a \). This provides an understanding of how to rearrange and solve logarithmic equations. For natural logarithms specifically, the base \(b\) is always \(e\), making it straightforward to apply these transformations.

• Knowing that \( \ln a = b \) can be rewritten as \(e^b = a\) helps in both solving problems and simplifying complex expressions.
• It opens different solving methods for equations that initially might seem daunting but get simplified through such manipulations.

These rules form the backbone of algebraic manipulation in many areas of mathematics and science.