How to Solve the Equation ln(x) = 2?

To solve the equation ln(x) = 2, the exact answer for x is e². This solution is derived by understanding the fundamental inverse relationship between the natural logarithm and the exponential function.

Understanding the Natural Logarithm (ln)

The natural logarithm, denoted as ln(x), is the inverse function of the exponential function e^x. In simpler terms, if ln(x) = y, it means that e raised to the power of y equals x (i.e., e^y = x). The constant e is an irrational number approximately equal to 2.71828.

It's crucial to remember that the natural logarithm ln(x) is only defined for positive values of x. This means that x must always be greater than zero.

Step-by-Step Solution to ln(x) = 2

Solving ln(x) = 2 involves a straightforward process using the inverse property of logarithms and exponentials.

Identify the Equation:
Your equation is ln(x) = 2. Here, ln(x) is already isolated on one side.
Exponentiate Both Sides:
To undo the natural logarithm, we apply the exponential function e to both sides of the equation. This is the key step that leverages the inverse relationship.
e^(ln(x)) = e^2
Simplify Using the Inverse Property:
The property of inverse functions states that e^(ln(x)) simplifies directly to x. This is because the exponential function and the natural logarithm cancel each other out.
x = e^2
Verify the Domain:
The solution x = e^2 is a positive number (since e itself is positive). This satisfies the domain requirement for ln(x), which states that x must be greater than 0.

Therefore, the exact solution to ln(x) = 2 is x = e^2. While e^2 can be approximated as 7.389, e^2 is the precise mathematical answer.

Key Concepts for Solving Logarithmic Equations

Understanding the core properties of logarithms and exponential functions is vital for solving such equations.

Concept	Description
Natural Logarithm (`ln(x)`)	Asks "to what power must `e` be raised to get `x`?".
Exponential Function (`e^x`)	The constant `e` raised to the power of `x`.
Inverse Property	`e^(ln(x)) = x` and `ln(e^x) = x`. These functions undo each other.
Domain Restriction	For `ln(x)`, `x` must always be greater than `0`.

Practical Insights

Exact vs. Approximate Answers: In mathematics, e^2 is considered the exact answer. An approximate decimal value (e.g., 7.389) should only be used if specified.
Generalization: This method applies to any equation of the form ln(x) = k, where k is a constant. The solution will always be x = e^k. For example, if ln(x) = 5, then x = e^5.