Problem 251 - Calculus Volume 1

Question

Write the equation in equivalent exponential form. \(\quad \ln \left(\frac{1}{e^{3}}\right)=-3\)

Step-by-Step Solution

Answer

The equivalent exponential form is \( e^{-3} = \frac{1}{e^{3}} \).

Step 1: Understand the Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \) (where \( e \approx 2.718 \)). If \( \ln(a) = b \), it implies \( e^b = a \).

Step 2: Identify the Given Expression

The equation given is \( \ln \left(\frac{1}{e^{3}}\right) = -3 \). This is in the form \( \ln(a) = b \), where \( a = \frac{1}{e^{3}} \) and \( b = -3 \).

Step 3: Convert to Exponential Form

Using the property of natural logarithms, we express the equation \( \ln \left(\frac{1}{e^{3}}\right) = -3 \) in exponential form. This becomes \( e^{-3} = \frac{1}{e^{3}} \).

Step 4: Verify the Conversion

Check if the exponential form \( e^{-3} = \frac{1}{e^{3}} \) holds true. This is correct because \( e^{-3} \) is equivalent to \( \frac{1}{e^{3}} \), confirming our conversion.

Key Concepts

Natural Logarithm

The natural logarithm, often represented as \( \ln(x) \), is a special type of logarithm where the base is the mathematical constant \( e \). This constant \( e \) is approximately equal to 2.71828 and is an irrational number that's important in many areas of mathematics, especially calculus and complex systems.
The natural logarithm serves as the inverse of the exponential function with base \( e \). Let's unwrap what this really means: if you have an equation like \( \ln(a) = b \), it's equivalent to saying \( e^b = a \).

• It's crucial to note that natural logarithms are only defined for positive real numbers.
• The value \( a \) in \( \ln(a) \) must be positive because you can't take the logarithm of a non-positive number in the real number system.

Understanding these principles helps bridge the gap between logarithmic and exponential equations, allowing one to switch effortlessly between the two forms.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. This means that they 'undo' what exponential functions do. If an exponential function takes a number as an exponent, a logarithmic function takes the result of that exponentiation and returns the initial number.
A logarithm answers the question: "To what exponent must the base be raised to yield a specific number?" This can be written as \( b^y = x \), where \( \log_b(x) = y \). For natural logarithms, the base is the number \( e \).

• Logarithmic functions decrease as their inputs go to zero and increase slowly for larger inputs.
• They are particularly useful in real-world applications such as measuring sound intensity (decibels), the Richter scale for earthquakes, and in financial contexts for calculating certain kinds of interest.

By using logarithmic properties, such as the one demonstrated in our exercise, you can transform complex logarithmic equations into more manageable forms, making complex calculations more accessible.

Exponential Form

Converting a logarithmic statement into exponential form can simplify the equation and make it easier to solve. For example, you can translate \( \ln(a) = b \) into the equivalent expression \( e^b = a \).
In the problem we worked on, \( \ln \left(\frac{1}{e^3}\right) = -3 \) was converted into the exponential form \( e^{-3} = \frac{1}{e^3} \). By doing this transformation, we find a direct equivalence between a logarithmic statement and an exponential one.

• This technique is highly advantageous when seeking to isolate variables or simplify expressions.
• It's frequently used in algebra and calculus to resolve equations involving rates of growth and decay, like compounding interest or radioactive decay.

Being able to switch between logarithmic and exponential forms is a pivotal skill in mathematics. It enriches your toolbox when dealing with real-world problems where exponential growth or decay patterns arise.