Problem 41 - Algebra and Trigonometry

Question

Solve the logarithmic equation for \(x\) $$ 2-\ln (3-x)=0 $$

Step-by-Step Solution

Answer

The solution is \( x = 3 - e^2 \).

Step 1: Isolate the Logarithmic Term

We start by isolating the logarithmic term. The given equation is: \[ 2 - \ln(3 - x) = 0 \]Subtract 2 from both sides to isolate the logarithmic term:\[ -\ln(3 - x) = -2 \]Now multiply both sides by -1 to get:\[ \ln(3 - x) = 2 \]

Step 2: Exponentiate to Remove the Logarithm

To eliminate the natural logarithm, we exponentiate both sides. Recall that if \( \ln(a) = b \), then \( a = e^b \).Here, apply this property:\[ 3 - x = e^2 \]

Step 3: Solve for x

Now, solve for \(x\) by isolating it on one side of the equation. Start with:\[ 3 - x = e^2 \]Subtract 3 from both sides to isolate \(-x\):\[ -x = e^2 - 3 \]Next, multiply both sides by -1 to solve for \(x\):\[ x = 3 - e^2 \]

Key Concepts

Natural Logarithm

The natural logarithm is a key mathematical concept denoted as \( \ln \), which stands for the logarithm to the base \( e \), where \( e \approx 2.71828 \). It's an essential tool in calculus and many other areas of mathematics and engineering. Understanding natural logarithms helps simplify complex exponential relationships by converting them into manageable forms.

Key properties of natural logarithms include:

• The natural logarithm of 1 is 0: \( \ln(1) = 0 \)
• The natural logarithm of \( e \) itself is 1: \( \ln(e) = 1 \)
• Natural logarithms obey several rules like product (\( \ln(ab) = \ln a + \ln b \)), quotient (\( \ln(a/b) = \ln a - \ln b \)), and power rules (\( \ln(a^b) = b \ln a \)).

When solving equations involving the natural logarithm, it is helpful to remember these rules, as they can be used to simplify the expressions and isolate variables.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. With logarithmic equations, like \( 2 - \ln(3-x) = 0 \), our goal is to isolate and solve for \( x \).

Here's the general plan:

• Start by isolating the logarithmic expression on one side. For our example, this means rearranging the equation to \( \ln(3-x) = 2 \).
• Use exponentiation to eliminate the logarithm, converting the logarithmic statement into an exponential form. According to the property that \( \ln(a) = b \) implies \( a = e^b \), we find \( 3 - x = e^2 \).
• Finally, solve for the variable \( x \) by continuing to manipulate the equation. Ensure \( x \) is by itself, resulting in \( x = 3 - e^2 \).

Following these steps, logarithmic equations become more straightforward, transforming intricate logs into simple algebraic expressions.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The operation involves raising the base to the power of the exponent,
which means multiplying the base by itself as many times as the exponent indicates. It is the inverse operation of logarithms.

In the context of solving logarithmic equations, exponentiation is a crucial step in removing the logarithm from the equation.

• For example, the equation \( \ln(3-x) = 2 \) can be converted to exponential form as \( 3-x = e^2 \) by raising \( e \) (the base of natural logs) to both sides of the equation.
• This step effectively "undoes" the logarithm, allowing us to solve for the unknown variable with standard algebraic techniques.

Understanding exponentiation as the reverse of logarithms not only helps in solving logarithmic equations but also reinforces the relationship between these two mathematical operations. It’s a powerful strategy for isolating variables and finding solutions in exponential growth and decay problems as well.