Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (2 x)=5$$

Step-by-Step Solution

Answer

\( x = \frac{e^5}{2} \), approximately 74.21

Step 1: Isolate the Logarithm

The equation is \( \ln(2x) = 5 \). There is already a logarithmic function isolated on one side. This step is completed since \( \ln(2x) \) stands alone.

Step 2: Exponentiate to Eliminate the Logarithm

To eliminate the natural logarithm, take the exponential of both sides. This gives \( e^{\ln(2x)} = e^5 \). Since \( e^{\ln(2x)} = 2x \), we have \( 2x = e^5 \).

Step 3: Solve for x

Now solve for \( x \) by dividing both sides by 2: \( x = \frac{e^5}{2} \).

Step 4: Use Calculator to Verify Solution

Using a calculator, compute \( e^5 \approx 148.413159 \) and verify \( \frac{e^5}{2} \approx 74.2065795 \). This supports the exact form solution, \( x = \frac{e^5}{2} \).

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a special type of logarithm where the base is the mathematical constant \( e \). The number \( e \) is approximately equal to 2.71828 and is an irrational number. The natural logarithm helps simplify calculations involving exponential growth and decay, making it particularly useful in fields like calculus and finance. Logarithms convert multiplication into addition, which simplifies analyses of growth patterns.
For example, the natural logarithm \( \ln(x) \) asks the question: "To what power must \( e \) be raised to get \( x \)?" This question is fundamental when dealing with equations where the growth rate is proportional to the current value, like in our original exercise.

The Role of Exponents in Solving Logarithmic Equations

Exponents are mathematical notations that describe repeated multiplication of a number by itself. In the context of solving logarithmic equations, exponents are crucial for eliminating the logarithm from an equation.
In our exercise \( \ln(2x) = 5 \), we took advantage of the property that \( e^{\ln(a)} = a \) to remove the logarithm. By taking the exponential of both sides of the equation, we converted the logarithmic form into an exponential form:

• \( e^{\ln(2x)} = e^5 \)

This simplifies to \( 2x = e^5 \). The skillful use of exponents opens the path to finding the value of \( x \), and highlights the relationship between logarithms and exponents as inverse operations.

Using Calculators for Verification

Calculators are essential tools for solving and verifying solutions to logarithmic equations, especially when dealing with complex numbers and exponential expressions. When you exponentiate to eliminate a natural logarithm, the numerical result often isn't an easy number to calculate mentally. That's where a calculator becomes invaluable.
In our example, after solving for \( x \), we needed to compute \( e^5 \) to verify our solution. Most scientific calculators have \( e \) and natural logarithm functions readily available, making it straightforward to check calculations.

• \( e^5 \approx 148.413159 \)

This calculation confirms our exact form solution, ensuring accuracy and reliability in our results.

Expressing Solutions in Exact Form

Exact form solutions are important in mathematics because they preserve all the precision of a mathematical expression without approximation. Unlike decimal approximations—which can lose accuracy—exact form expressions convey the complete mathematical information using symbols like \( e \) or \( \pi \).
In solving \( \ln(2x) = 5 \), we expressed the solution as \( x = \frac{e^5}{2} \). This expression is an exact form, capturing the precise mathematical relationship without losing any detail to rounding or estimation. Using exact forms often makes subsequent calculations and proofs much cleaner and more accurate.