Problem 65 - Algebra 2

Question

Solve each equation. $$ \ln x^{2}+1=5 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = e^{2}\) and \(x = -e^{2}\)

Step 1: Isolate the Logarithmic Expression

The first step is to isolate the logarithmic term on the left-hand side, to do so we must subtract 1 from both sides. This will give us: \(ln(x^{2}) = 5 - 1 = 4\)

Step 2: Convert Logarithm to Exponential Form

To get rid of the natural logarithm ('ln'), we can rewrite the equation in an exponential form. This results in \(x^{2} = e^{4}\)

Step 3: Solve for x

The final step is to solve for x. By taking the square root of both sides we get: \(x = \pm \sqrt{e^{4}} = \pm e^{2}\)

Key Concepts

Natural Logarithm

The concept of a natural logarithm is an essential building block in understanding logarithmic equations. The natural logarithm, often represented as \(\ln\), has a base of \(e\), which is approximately equal to 2.718. When we deal with \(\ln\), it's important to remember that it is the inverse operation to taking the exponential function of base \(e\).

In simpler terms, if you take the natural logarithm of a number \(x\), you are essentially asking: **"To what power must \(e\) be raised to result in \(x\)?**"

So, if \(\ln(x) = 4\), it implies that \(e\) raised to the power of 4 equals \(x\).

Natural logarithms are incredibly useful because they allow us to transform exponential relationships into linear ones, making them easier to handle in mathematics. They are widely used, for example, in calculating continuous growth rates in finance and biology.

Exponential Equations

Exponential equations are equations where variables appear as exponents. In solving such equations, transforming the expression using logarithms, especially the natural logarithm, is often a key strategy.

When you encounter an equation like \(x^2 = e^4\), the base \(e\) makes the problem an exponential equation. Exponential forms can look complex, but they can be broken down into understandable steps:

• First, express the equation using a logarithm to simplify.
• Use the property of logarithms, stating that both sides of an equation can be transformed to logarithmic terms, to find equivalent solutions.
• With equations of similar kinds \(a^x = b\), it allows us to solve for \(x\) by taking logarithms on both sides.

In our previous exercise, we did just that. By rewriting \(\ln(x^2)\) as an exponential equation \(x^2 = e^4\), we managed to turn a logarithmic question into one solvable by taking square roots.

Solving Equations

Solving equations can often seem daunting, but with a structured approach, it becomes manageable. Solving equations typically involves finding the value of the variable that makes the equation true. With logarithmic equations like \(\ln(x^2) + 1 = 5\), the key steps include isolating the logarithmic term, converting to exponential form, and solving for the variable.

Here's a simple strategy to follow when solving equations, especially those involving logarithms and exponents:

• Isolate the logarithmic expression: Ensuring the logarithmic part stands alone helps clarify the next steps.
• Convert to exponential form: By rewriting the logarithmic equation in exponential terms, it becomes simpler.
• Solve for the variable: After converting, apply algebraic methods, like taking roots, to find the variable's value.

This structured approach ensures that each step logically follows the previous one. Whether dealing with natural logarithms or other forms, isolating, converting, and solving are consistent methods to reach the right solution quickly.