Question

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ e^{-3}=0.0498 \ldots $$

Short answer

The logarithmic form of the given equation is \( \ln{0.0498} = -3 \)

Step-by-step solution

Recognize the Exponential Form

The given equation is in the exponential form: \( e^{-3} = 0.0498 \)

Write as Logarithmic Equation

The logarithmic form of the above equation is obtained by recognizing that an exponentiation equation, \( b^x = y \), is equivalent to a logarithmic equation \( \log_b{y} = x \). So, the logarithmic form of the given equation is: \( \log_e{0.0498} = -3 \)

Final Simplification

The logarithm base \( e \) can be written as \( \ln \), so the final simplified form of the equation is: \( \ln{0.0498} = -3 \)

Key concepts

Exponential Equations

Exponential equations are mathematical expressions where a variable appears in the exponent rather than a typical position as a multiplier or an addend. These types of equations are prevalent in fields like science, finance, and other areas where growth or decay processes need to be modeled. For example, population growth or radioactive decay often use exponential equations to describe their processes.
In the given exercise, we start with the exponential equation \( e^{-3} = 0.0498 \). Here, \( e \) is the base of the natural logarithm, also known as Euler's number, approximately equal to 2.71828. Exponential equations can often look foreign at first due to the use of constants like \( e \), but the principle is straightforward: the exponent \(-3\) indicates the power to which \( e \) must be raised to yield 0.0498.

• **Key Aspect 1**: An exponential equation in the form \( b^x = y \) denotes that base \( b \) raised to the power of \( x \) gives \( y \).
• **Key Aspect 2**: Typically, solving exponential equations may involve finding a common base, isolating the exponent, or converting into logarithmic form, as discussed in the exercise.
• **Key Aspect 3**: Exponential growth occurs when a quantity increases rapidly over time; exponential decay refers to the quantity decreasing, such as with a negative exponent.

Understanding these characteristics allows you to maneuver problems efficiently using exponential equations.

Logarithmic Form

Logarithmic form fundamentally rearranges an exponential equation into an expression where the focus is on finding that exponent. In simpler terms, if you have an exponential equation such as \( b^x = y \), then the equivalent logarithmic form is written as \( \log_b{y} = x \). Think of logarithms as the inverse operation to exponentiation.
In our specific exercise, the conversion from the exponential form \( e^{-3} = 0.0498 \) into logarithmic form is made possible by recognizing the inverse nature of logarithms. We rearrange it to \( \log_e{0.0498} = -3 \), strongly emphasizing that \(-3\) is the power to which the base \( e \) was raised.

• **Key Concept 1**: Understanding the relationship between exponentials and logarithms is crucial; they're different ways of expressing the same mathematical relationship.
• **Key Concept 2**: When transitioning from exponential to logarithmic form, the original exponent becomes the result of the logarithm.
• **Key Concept 3**: A natural logarithm uses \( e \) as its base and is often denoted as \( \ln \), which simplifies math notation and computation processes.

Overall, the logarithmic form provides a dynamic tool in isolating variables, especially within exponential contexts.

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a logarithm with the base \( e \), where \( e \approx 2.71828 \). It's a crucial tool in calculus and many fields of science and engineering. Its unique properties allow for seamless integration and differentiation, making it essential in both theoretical and applied mathematics.
In the given exercise, the final step involves recognizing that \( \log_e{0.0498} \) can be represented simplistically as \( \ln{0.0498} \). This not only emphasizes the prevalence of \( e \) in natural growth and decay processes but pinpoints \(-3\) as the exponent.

• **Critical Insight 1**: Natural logs simplify equations by eliminating cumbersome base notation and are standardized in many mathematical formulations.
• **Critical Insight 2**: They emerge naturally in scenarios involving continuous growth or decay, such as calculating compounded interest or understanding biochemical reaction rates.
• **Critical Insight 3**: Every natural logarithm \( \ln{x} \) solves the equation \( e^y = x \), showcasing its strong ties to exponential functions.

Grasping the importance of natural logarithms equips you with a powerful mathematical tool for dealing with real-world and theoretical problems alike.