Problem 61 - 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. $$ 5 \ln (2 x)=20 $$

Step-by-Step Solution

Answer

The solution for \(x\) is \(e^4 / 2\) and in decimal form it's '27.47'.

Step 1: Division of Both Sides of the Equation

Divide all terms of the equation by the coefficient of \(\ln (2 x)\), which is '5', to isolate the logarithmic expression on one side. It gives \(\ln (2x) = 4\).

Step 2: Conversion to Exponential Form

Convert the logarithmic equation to an exponential equation. The base of logarithm is 'e', thus, the equation transforms to: \(e^4 = 2x \).

Step 3: Solve for x

Divide both sides of the equation by '2' to get the value of \(x\). So, the solution for \(x\) is \(x = e^4 / 2\).

Step 4: Decimal Approximation

The exact solution for \(x\) is \(e^4 / 2\). Using a calculator, the decimal approximation value for \(x\) up to two decimal places will be '27.47'. Finally, since the logarithmic function \(\ln(2x)\) is undefined for \(x \leq 0\), check the solution and confirm that it satisfies \(x > 0\). This solution \(x = 27.47\) is a valid solution because it is greater than zero.

Key Concepts

Domain of Logarithmic Functions

When dealing with logarithmic equations, it's crucial to understand the domain of the logarithmic functions. The domain is the set of input values (here, the variable values) for which the function is defined. For the natural logarithm function, written as \( \ln(x) \), the domain consists of all positive real numbers. This means that \( x \) must be greater than zero in order for \( \ln(x) \) to be defined.In our provided exercise, the expression inside the logarithm is \( 2x \). Consequently, the condition becomes \( 2x > 0 \). Simplifying this inequality, it shows that \( x > 0 \). Therefore, any solution for \( x \) must satisfy this condition in order to be valid. If it doesn't, the solution will be rejected because it leads to a logarithmic value that is undefined.

Exponential Form

Converting a logarithmic equation into its exponential form is an essential skill. This transformation helps in solving logarithmic equations more easily. The general rule is that if you have a logarithm of a certain base, say \( a^b = c \), then the equivalent logarithmic form is \( \log_a(c) = b \).In the given problem, we transform from \( \ln(2x) = 4 \) to its exponential equivalent. Since the base of the natural logarithm \( \ln \) is \( e \), the equation \( \ln(2x) = 4 \) converts to \( e^4 = 2x \). This transformation simplifies the equation, allowing us to solve for \( x \) by simple algebraic manipulation.

Decimal Approximation

Exact solutions are wonderful in theory, but in practice, we often need to work with decimal approximations. This is especially true when dealing with irrational numbers like \( e \). A decimal approximation provides a more tangible result that can be easily utilized in practical applications.In our exercise, after converting to exponential form and isolating \( x \), we end up with \( x = e^4 / 2 \). Using a calculator, we approximate \( e^4 \) first and then divide by 2 to get \( x \approx 27.47 \). By providing the value to two decimal places, the solution becomes not only exact for further calculation but also practical and straightforward to read.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special logarithm with the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. It's one of the most common logarithms used, particularly in calculus and higher mathematics, because it has unique properties that simplify the product, quotient, and power rules.Understanding \( \ln \) is crucial not only for solving logarithmic equations but also for appreciating its applications in real-world processes such as exponential growth and decay, which occur naturally. In our given exercise, \( \ln(2x) = 4 \) means that the growth level \( e^4 \) equates to \( 2x \). This transformation allowed us to isolate and find the value of \( x \). Recognizing how natural logarithms connect to exponential relationships is pivotal for grasping many scientific and engineering principles.