Problem 33 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$5 \ln x=10$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

We start with the equation: \(5 \ln x = 10\). The goal is to isolate \(x\).

Step 2: Divide by 5

To isolate the natural logarithm, divide both sides of the equation by 5: \(\ln x = \frac{10}{5}\). This simplifies to \(\ln x = 2\).

Step 3: Exponentiate Both Sides

To solve for \(x\), you exponentiate both sides of the equation to eliminate the natural logarithm. Write it as: \(e^{\ln x} = e^2\). Since \(e^{\ln x} = x\), we have \(x = e^2\).

Step 4: Use a Calculator for Verification

Use a calculator to verify the solution. Calculate \(e^2\) using the "e^x" function on a scientific calculator, which should result in approximately \(x \approx 7.389\).

Step 5: Confirm the Exact Form

The problem asks for the exact form. We have found that \(x = e^2\) is the exact form solution, and the calculator calculation shows it's accurate.

Key Concepts

Natural Logarithm

Logarithms, especially natural logarithms, are foundational in mathematics. The natural logarithm, denoted as \(\ln x\), is a special logarithm with a base of \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. It is used frequently in calculus, especially when dealing with growth processes or compound interest. 

• Natural logarithms help us transform exponential equations into a linear form which can be easier to solve.
• The inverse of the natural logarithm function is the exponential function, \(e^x\).

In the given equation \(5 \ln x = 10\), the natural logarithm allows us to isolate \(x\) by converting multiplicative factors of logarithms into powers, making it easier to solve for the variable.

Exponentiation

Exponentiation is the process of raising a number to the power of another. In the context of solving logarithmic equations, it serves as the inverse operation to logarithms.

• When you have an equation involving a natural logarithm, like \(\ln x = 2\), you can solve for \(x\) by exponentiating both sides using the mathematical constant \(e\) as the base.
• Exponentiation cancels out the natural logarithm. For example, if you exponentiate both sides of \(\ln x = 2\), you get \(e^{\ln x} = e^2\).

Consequently, the exponential and logarithmic functions effectively "undo" each other, allowing for the isolation of \(x\), providing \(x = e^2\) in this case.

Exact Form Solutions

Exact form solutions are representations of answers without approximations or rounding. In mathematics, expressing solutions in exact form often involves symbols like \(e\), \(\pi\), or square roots. 

• These solutions are important as they convey the precision of mathematical relationships.
• They do not rely on decimals which can approximate but not fully capture the exactness of certain numbers.

In our exercise, the solution is expressed in exact form as \(x = e^2\). This form of solution is crucial in situations where precise mathematical manipulation is required or when further algebraic operations might be needed.
The exact forms serve as a clear, concise representation of the solution, establishing a clear understanding of the underlying mathematical principles without depending on a numerical approximation.