Problem 36 - Algebra and Trigonometry

Question

Solve the logarithmic equation for \(x\) $$ \ln (2+x)=1 $$

Step-by-Step Solution

Answer

The solution is \( x = e - 2 \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \ln (2+x) = 1 \). This is a natural logarithm equation, where \( \ln \) refers to the natural logarithm with base \( e \). Our goal is to solve for \( x \).

Step 2: Exponentiate Both Sides

To remove the logarithm, exponentiate both sides using base \( e \). This will give us: \[ e^{\ln (2+x)} = e^1 \]. Since \( e^{\ln (a)} = a \), we have \( 2+x = e \).

Step 3: Solve for x

Now that the equation is simplified to \( 2+x = e \), solve for \( x \) by isolating it on one side: \[ x = e - 2 \].

Key Concepts

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a logarithm with the base \( e \). The number \( e \) is a mathematical constant approximately equal to 2.71828, and it's often referred to as Euler's number. Natural logarithms are widely used in mathematics, especially when dealing with exponential functions. When you see \( \ln(y) \), it's asking the question: "To what power must \( e \) be raised to produce \( y \)?"
\( \ln \) is particularly useful because it simplifies differential calculus, growth processes, and compound interest in finance.

• For example, in the equation \( \ln(2+x) = 1 \), the natural logarithm lets us relate addition inside the logarithm to an exponent outside of it.
• The key property is that \( e^{\ln(a)} = a \), which allows us to easily "undo" the logarithm by exponentiating.

Understanding how \( \ln \) interacts with exponential terms is essential for solving equations like the one in our exercise.

Exponentiation

Exponentiation involves raising a number, known as the base, to a specific power or exponent. In our context, we use exponentiation with the base \( e \), which plays a critical role when working with natural logarithms. By exponentiating, we "cancel out" the natural logarithm.
In the given exercise, we start with the equation \( \ln(2+x) = 1 \). To "remove" the logarithm, we raise both sides as powers of \( e \), like this:

• \( e^{\ln(2+x)} = e^1 \).

This transformation leverages the property \( e^{\ln(a)} = a \), effectively simplifying \( \ln(2+x) \) to \( 2+x \). This step brings us closer to solving the equation because the equation \( 2+x = e \) has only one variable.
Exponentiating is a powerful method for transforming logarithmic equations into simpler ones that can be solved with traditional algebraic techniques.

Solving Equations

Equations often require a series of manipulations to isolate the variable you're solving for.
The process typically involves simplifying expressions, using inverse operations, and re-arranging the equation until the desired variable stands alone.

• With the equation \( 2+x = e \), solving for \( x \) means isolating \( x \) on one side of the equation.

To do this, we simply subtract 2 from both sides, resulting in:

• \( x = e - 2 \).

This straightforward algebraic manipulation finalizes the solution, giving us \( x \) in terms of \( e \). Recognizing steps like these can simplify even the most complex equations. Remember, the goal is always to isolate the variable by performing operations that "undo" each other, getting you closer to the solution.