Problem 45 - Precalculus with Limits

Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. \(\ln x=-3\)

Step-by-Step Solution

Answer

The solution to the equation \(\ln x = -3\) is \(x = e^{-3}\), which approximates to 0.050 when rounded to three decimal places.

Step 1: Understanding logarithms

Comprehend the nature of logarithms. In this equation, \(\ln x = -3\), the term \(\ln x\) means 'the power to which a base of e (approximately 2.71828) must be raised to equal x'. So, the equation can be rewritten in exponential form.

Step 2: Converting logarithmic equation into an exponential equation

Convert the logarithmic equation \(\ln x = -3\) into its equivalent exponential form. Using the inverse property of logarithms, the equation becomes \(e^{-3} = x\).

Step 3: Calculating value of x

Calculate the value of x by finding exponentiation of e to the power of -3. Here you also need to round the value of x to three decimal places.

Key Concepts

Understanding Exponential Equations

Exponential equations involve expressions that have a constant raised to the power of a variable. In our example, when converting the logarithmic equation \( \ln x = -3 \) into exponential form, it becomes \( e^{-3} = x \). Here, \( e \) is the base of the natural logarithm, approximately 2.71828. This makes \( e^{-3} \) an exponential expression.

The key points to understand about exponential equations include:

• They show how rapidly numbers can grow or shrink with a base raised to varying powers.
• The base, \( e \), is a fundamental constant used in natural logarithms and arises frequently in calculus and real-world applications.
• Exponential equations can often be solved using logarithms by isolating the exponential expression and then taking the logarithm of both sides.

Simplifying such equations involves applying inverse operations and properties to find the unknown variable.

Demystifying the Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a specific type of logarithm that uses \( e \) as its base. The expression \( \ln x = y \) means that \( x \) is \( e \) raised to the power \( y \).

Key characteristics of the natural logarithm:

• The natural logarithm is used to model growth or decay processes that occur continuously, such as populations or radioactive decay.
• Natural logarithms are the inverse functions of exponential functions with base \( e \). This means they "undo" exponentiation with base \( e \).
• They are widely used in scientific fields due to their unique mathematical properties and relevance to growth processes.

Understand that the natural log transforms multiplicative relationships into additive ones, making complex calculations more manageable.

Inverse Properties in Solving Equations

Inverse properties are essential tools in algebra. They help us solve equations by reversing operations. For example, exponentiation and logarithms are inverse operations. They cancel each other out.

Here’s how inverse properties work:

• If you have \( \ln(x) \), applying the exponential function with base \( e \) (denoted \( e^{\ln(x)} \)) gives you back \( x \), as they are inverses.
• Similarly, if you have \( e^x \), applying the natural log (\( \ln(e^x) \)) returns \( x \).

This relationship is crucial in converting logarithmic equations to exponential form and vice versa, which is precisely what we did in our example problem. Using inverse properties, we took \( \ln x = -3 \) and transformed it into \( e^{-3} = x \), enabling us to find \( x \) directly by calculating the exponential value.