Problem 59 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln e^{x}-2 \ln e=\ln e^{4}$$

Step-by-Step Solution

Answer

The solution is \( x = 6 \).

Step 1: Apply the Logarithmic Identity

We start by recalling that \ \( \ln(a^b) = b \ln(a) \). Hence, we can simplify the term \( \ln(e^x) \) to \( x \ln(e) \). Since \( \ln(e) = 1 \), this becomes \( x \).

Step 2: Simplify the Equation

Using the identity from Step 1, rewrite the equation \( \ln(e^x) - 2 \ln(e) = \ln(e^4) \) to \( x - 2 = 4 \).

Step 3: Solve for x

Add 2 to both sides of the equation \( x - 2 = 4 \) to solve for \( x \). We get \( x = 4 + 2 \). Thus, \( x = 6 \).

Step 4: Verify the Solution

Substitute \( x = 6 \) back into the original equation to verify: \( \ln(e^6) - 2 \ln(e) = \ln(e^4) \). The left side simplifies to \( 6 - 2 \times 1 = 4 \), confirming that the two sides are equal. The solution \( x = 6 \) is correct.

Key Concepts

Logarithmic Identities

Logarithmic identities are powerful tools used in simplifying and solving logarithmic equations. One key identity is the power rule: \( \ln(a^b) = b \ln(a) \). This means that the logarithm of an exponentiated number can be rewritten as a product of the exponent and the logarithm of the base.
For example, in the equation \( \ln(e^x) \), applying the power rule allows us to simplify this to \( x \cdot \ln(e) \). Since \( \ln(e) = 1 \), this expression further simplifies to \( x \).
It's crucial to understand these identities, as they help transform and solve complex logarithmic equations. Knowing when and how to apply each identity can turn an intricate equation into a much simpler form, ultimately helping in finding solutions efficiently.

Solving Logarithms

Solving logarithmic equations involves manipulating the equation to isolate the variable. Let's break it down step-by-step.
First, utilize the power rule of logarithms to simplify the expressions. In our example, \( \ln(e^x) \) becomes \( x \). This step reduces complexity and brings us closer to isolating \( x \).
Next, aim to rearrange the equation to have the variable on one side. For instance, transforming the equation \( x - 2 = 4 \) helps in directly solving for \( x \). Finally, perform arithmetic operations to solve for the unknown variable. Here, adding 2 to both sides gives \( x = 6 \).

• Apply logarithmic identities to simplify logs.
• Rearrange to isolate the variable.
• Perform necessary arithmetic operations for solution.

By following these steps, solving logarithmic equations becomes systematic and straightforward.

Exact Solutions

In mathematics, exact solutions refer to solutions that are expressed precisely, without approximations. When solving logarithmic equations, expressing solutions in exact form is crucial for accuracy and understanding. This means avoiding decimal approximations and retaining mathematical expressions as they are.
In our exercise, the solution \( x = 6 \) is an exact value derived from a series of logical and arithmetic steps, remaining faithful to the original terms of the equation.
Emphasizing exactness is important, especially in contexts requiring precise calculations, such as engineering or physical sciences. It ensures consistency and dependability in results, making it a valuable practice in problem-solving.

Verification of Solutions

Verifying solutions is a critical step in problem-solving, ensuring that the obtained answers are correct. After solving a logarithmic equation, it's vital to substitute the solution back into the original equation to check its validity.
In our example, substituting \( x = 6 \) into the equation \( \ln(e^x) - 2 \ln(e) = \ln(e^4) \) results in \( 6 - 2 = 4 \). Since both sides of the equation are equal, the solution is verified as correct.
This verification step is not just a formality; it provides assurance that no errors were made during the solving process. It's a good practice to develop as it reinforces the reliability of your solutions. Plus, it boosts confidence in your mathematical abilities. Before moving on to another problem, always verify your solution to ensure accuracy.