91Ó°ÊÓ

The natural logarithm, denoted as \(\text{ln}\), is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. Unlike regular logarithms, which can have any positive number as their base, the natural logarithm always uses \(e\) as its base.
We use the natural logarithm when dealing with exponential growth or decay, especially in the context of solving equations where the unknown variable is in the exponent.
When you take the natural logarithm of \(e\) raised to any power, \(y\), the natural log and the exponential function cancel out, leaving just \(y\). That means \( \text{ln}(e^y) = y \).
For example, in the equation \(e^{2x} = 5\), we take \( \text{ln}(e^{2x}) \) to both sides to bring down the exponent 2x, which simplifies further using the logarithm property.

Logarithms have several important properties that can help simplify complex equations, making them easier to solve. Here are a few key properties:

• The Power Rule: \( \text{ln}(a^b) = b \text{ln}(a) \) which allows us to bring down exponents as coefficients.
  
• The Product Rule: \( \text{ln}(ab) = \text{ln}(a) + \text{ln}(b) \) which lets us separate the logarithm of a product into a sum for easier manipulation.
  
• The Quotient Rule: \( \text{ln}\bigg(\frac{a}{b}\bigg) = \text{ln}(a) - \text{ln}(b) \) which splits the logarithm of a quotient into a difference, aiding even further simplification.
  

Applying these properties in mathematical problems, like our given equation \(e^{2x} = 5\), is essential to isolate and solve for variables that are present in exponents.

Isolating a variable means rearranging an equation such that you have the variable by itself on one side of the equation. This is often a key step in solving equations.
When dealing with exponential equations like \(e^{2x} = 5\), after applying the natural logarithm to both sides, you're left with \( 2x = \text{ln}(5) \).
The next step is to solve for \(x\) by isolating it. This generally involves basic algebraic operations such as:

• Adding or subtracting terms from both sides.
  
• Multiplying or dividing both sides by a number or a variable.
  

For our equation, we divide both sides by 2, giving us \( x = \frac{\text{ln}(5)}{2} \).
Through these steps, isolating the variable helps us find the exact value of \(x\), showing the solution to the problem.