Problem 4 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ \ln 0.05=-2.9957 \ldots $$

Step-by-Step Solution

Answer

The logarithmic equation \( \ln 0.05 = -2.9957 \) can be rewritten in an equivalent exponential form as \( e^{-2.9957} = 0.05 \).

Step 1: Understand the relationship between logarithm and exponential form

Write down the logarithmic equation as follows - \( \ln 0.05 = -2.9957 \). In a logarithmic equation of the form \( \ln b = a \), b is the argument of the logarithm, a is the value of the logarithm, and the number e (approximately equal to 2.71) is the base.

Step 2: Convert the logarithm base

Since the natural logarithm \( \ln \) has base e, convert the base of the logarithm to e. The logarithmic equation becomes \( \log_{e} 0.05 = -2.9957 \). Now the equation reads 'base e raised to the power of -2.9957 is equal to 0.05'.

Step 3: Rewrite in exponential form

Since any logarithmic equation \( \log_{b} a = c \) can be written in exponential form as \( b^c = a \), rewrite the equation from step 2 in exponential form. Therefore, the equivalent exponential equation is \( e^{-2.9957} = 0.05 \).

Key Concepts

Natural Logarithm

The natural logarithm is a logarithmic function, where the base is the mathematical constant \( e \) (approximately equal to 2.71). It is denoted as \( \ln \) and is widely used in various mathematical calculations. A logarithm answers the question: "To what power must the base be raised, to produce a given number?" When discussing natural logs:

• The notation \( \ln b = a \) essentially says that base \( e \) raised to the power of \( a \) equals \( b \).
• \( \ln \) is often used in growth processes, such as population growth or decay processes like radioactive decay.
• Understanding natural logarithms is crucial in fields such as calculus, where they are used to solve exponential growth problems.

Natural logs simplify many complex calculations involving \( e \), making them essential tools for students and professionals alike. A key aspect is realizing that in a logarithmic equation, it's the power that becomes the central focus.

Exponential Form

The exponential form is a way of expressing equations where a base is raised to a power. Converting logarithmic equations into exponential form involves identifying the base and the relationship between the logarithm and the exponent:

• For any equation \( \ln b = a \), you can express it in exponential form as \( e^a = b \).
• This transformation is useful because it provides an intuitive way to understand growth rates and multiplicative processes.
• In the context of the original exercise, it demonstrates how the power \( -2.9957 \) when applied to base \( e \), results in 0.05, as in \( e^{-2.9957} = 0.05 \).

Switching to exponential form helps in visualization and can simplify calculations, especially when solving or graphing exponential equations. It forms the basis for understanding complex exponential relationships in algebra and beyond.

Base e

The base \( e \) is a fundamental mathematical constant approximately equal to 2.71. It appears frequently in mathematics due to its unique properties:

• It's the base of the natural logarithm, making it essential for calculations in both mathematics and natural sciences.
• \( e \) is an irrational number, meaning it cannot be expressed as a simple fraction and has an infinite decimal sequence without repeating.
• It's notable for its role in compound interest calculations and various natural phenomena, such as growth and decay processes.

Understanding base \( e \) is crucial in fields like calculus and differential equations because it simplifies the derivative of exponential functions. While it may seem abstract at first, recognizing the prevalence and importance of \( e \) in natural processes highlights its relevance in both theoretical and practical applications.