Problem 36 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \( x = \frac{e^5}{2} \).

Step 1: Understand the Equation

The given equation is \( \ln(2x) = 5 \). This is a natural logarithm equation where \( \ln \) denotes the natural logarithm function, which is the logarithm to the base \( e \). We need to solve for \( x \).

Step 2: Exponentiate Both Sides

To eliminate the natural logarithm, we exponentiate both sides of the equation. This means we raise \( e \) to the power of both sides of the equation: \[ e^{\ln(2x)} = e^5 \]. Since \( e^{\ln a} = a \), we simplify the equation on the left to \( 2x \).

Step 3: Simplify the Expression

The equation now becomes \( 2x = e^5 \). This is a simpler expression that can be solved for \( x \).

Step 4: Isolate x

To solve for \( x \), divide both sides by 2: \[ x = \frac{e^5}{2} \]. This is the exact form of the solution for \( x \).

Step 5: Verify with Calculator

Using a calculator, evaluate \( e^5 \) to get a numerical approximation. For instance, \( e^5 \approx 148.4132 \). Divide this by 2 to approximately confirm our solution: \( x \approx 74.2066 \).

Key Concepts

Natural Logarithm

The natural logarithm is denoted by \( \ln \) and is the logarithm with base \( e \), where \( e \) is approximately 2.718. The natural logarithm is a fundamental concept in mathematics, especially useful in calculus and complex analysis. Logarithms convert multiplicative relationships into additive ones, simplifying many calculations.

• When you see \( \ln(x) \), it specifically means the power to which \( e \) must be raised to produce \( x \).
• It is often used in solving equations that involve exponential growth or decay, such as in compound interest, population growth models, and radioactive decay.
• Understanding \( \ln \) helps simplify processes like solving equations that contain logarithmic expressions.

In the provided equation \( \ln(2x) = 5 \), the natural logarithm plays a role in making complex exponential problems manageable by helping to move equations from a logarithmic form to an exponential form, like \( e^{\ln(2x)} \).

Exponential Functions

Exponential functions are functions of the form \( f(x) = a^x \), where \( a \) is a positive real number and \( x \) is any real number. The most notable exponential function is \( e^x \), where \( e \approx 2.718 \), a crucial number in mathematics known as Euler's number.

• Exponential functions are characterized by rapid increases or decreases as the variable \( x \) changes.
• These functions are fundamental in modeling situations where change accelerates or decelerates at a constant rate, like population growth, interest in finance, and carbon dating.

The process of exponentiating both sides of the equation \( \ln(2x) = 5 \) involves using the exponential function \( e^x \). This operation reverses the logarithm, as \( e^{\ln a} = a \), thus helping to transform our problem from a logarithmic equation to a more straightforward linear form, \( 2x = e^5 \). This simplification is essential for finding the solution efficiently.

Exact Solutions

When solving equations, an exact solution provides a precise answer without approximations, typically expressed using constants and mathematical operations like addition, multiplication, exponents, and roots.

• In the equation \( 2x = e^5 \), finding the exact solution involves expressing \( x \) in a mathematically precise form rather than a numeric approximation.
• Exact solutions are especially important in theoretical math and professional contexts where precision is essential, such as engineering and physics.
• Even though approximations like \( x \approx 74.2066 \) are helpful for certain real-world applications, they lack the precision of the exact expression \( x = \frac{e^5}{2} \).

Emphasizing exact solutions ensures that mathematical integrity is maintained and that the foundational principles remain intact throughout the problem-solving process.