Problem 52 - Algebra and Trigonometry

Question

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \ln x=3 $$

Step-by-Step Solution

Answer

The exact solution is \(x = e^{3}\) and the decimal approximation to two decimal places is \(x = 20.09\).

Step 1: Understanding the properties of natural logarithm

The natural logarithm \(\ln x\) is the inverse operation of the exponential function with base \(e\) (Euler's number, approximately equal to 2.71828). Hence if \(\ln x = 3\), the operation can be rewritten in exponential form as \(e^{3} = x\).

Step 2: Solving for x

Rearranging the equation gives us \(x = e^{3}\). This is an exact solution.

Step 3: Calculating decimal approximation

If a decimal approximation is required to two decimal places, then using a calculator to evaluate \(e^{3}\) will give \(x \approx 20.09\).

Key Concepts

Natural Logarithm

The natural logarithm, denoted by \( \ln x \), is a logarithmic function that uses \( e \), Euler's number, as its base. The essence of a natural logarithm is to determine the power or exponent to which \( e \) must be raised to obtain a given number. For example, if \( \ln x = 3 \), this expression asks: "To what power must \( e \) be raised to equal \( x \)?"
The natural logarithm is fundamentally important in mathematics and science due to its appearance in various natural processes and mathematical modeling. Here are some key points to understand natural logarithms:

• The operation \( \ln x \) is only defined for \( x > 0 \).
• The inverse of \( \ln x \) is the exponential function, \( e^y = x \), which we will explore in the next section.
• Natural logarithms grow more slowly than any power of \( x \), which makes them useful in characterizing processes that decelerate over time.

Understanding the properties and applications of \( \ln x \) helps in solving equations where natural logarithms are involved. In our problem, knowing that \( \ln x = 3 \) translates into the exponential form supports the process of finding the solution.

Exponential Function

An exponential function can be recognized in mathematics as a process where a quantity increases by a constant factor per unit interval. It is expressed as \( f(x) = a^{x} \). In the context of natural logarithms, the exponential function involved uses the base \( e \): the specific form is \( e^x \).
When we say \( x = e^{3} \), we mean \( x \) is the number which results when \( e \) is raised to the power of 3. The properties of exponential functions include:

• For \( e^x \), any real number \( x \), the value of \( e^x \) is always positive.
• The exponential function grows at a rate proportional to its current value, which leads to its rapid increase.
• It is continuous and differentiable everywhere.

Exponential functions have numerous applications, such as in modeling population growth, radioactive decay, or interest calculations. By converting \( \ln x = 3 \) to its equivalent exponential form \( x = e^{3} \), we leverage the known properties of exponential functions to find the exact value of \( x \).

Decimal Approximation Calculation

Decimal approximation is a method used to find a number that is close to a given complex number. This is particularly useful when the exact value is difficult or impossible to work with, often because it is irrational like \( e^3 \). In the context of this exercise, the exact solution for \( x \) is \( e^3 \), a non-terminating, non-repeating decimal. Using a calculator helps us round this to a reasonable approximation.
To approximate \( e^3 \), usually a calculator or computational equipment calculates this and rounds it to two decimal places. The essential steps are:

• Input the value 3 into the exponential function with base \( e \) using a calculator.
• Observe the output, which is approximately 20.08553692.
• Round this to two decimal places, yielding an approximate value of \( 20.09 \).

This representation is vital in practical situations where a more user-friendly or understandable number is required for reporting or calculation purposes. Remember that the decimal approximation is not exactly the true value but practical for real-world uses.