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Logarithmic properties are rules that make it easier to work with logarithmic functions. They help simplify complex expressions and solve exponential equations. There are three main properties to focus on:

• Product Property: \(\text{ln}(ab) = \text{ln}(a) + \text{ln}(b)\)
• Quotient Property: \(\text{ln} \frac{a}{b} = \text{ln}(a) - \text{ln}(b)\)
• Power Property: \(\text{ln}(a^b) = b \text{ln}(a)\)

In this exercise, we specifically use the quotient property to rewrite the logarithmic function. This property states that the logarithm of a quotient is the difference of the logarithms. For \(Y_2 = \text{ln} x - \text{ln} 2\), we use the quotient property to transform it into \(Y_2 = \text{ln}\frac{x}{2}\). Understanding these properties is crucial for working with logarithmic functions efficiently.

Logarithmic functions are the inverse of exponential functions. They are of the form \(y = \text{ln}(x)\), where \(y\) is the logarithm of \(x\) to the base \(e\) (Euler's number). This means if \(y = \text{ln}(x)\), then \(e^y = x\). Here are some key points about logarithmic functions:

• The domain is \(x>0\).
• The range is all real numbers.
• The graph passes through (1, 0) because \( \text{ln}(1) = 0 \).

In the exercise, \(Y_1\) and \(Y_2\) are both logarithmic functions, where \(Y_1 = \text{ln} \frac{x}{2}\) and \(Y_2 = \text{ln} x - \text{ln} 2\). The properties of logarithms make them easier to compare and simplify.

Graphing logarithmic functions involves plotting points that satisfy the equation. Here's how to proceed:
1. Create a table of values for \(x\) and compute corresponding \(y\) values.
2. Plot these points on a coordinate plane.
3. Connect the points with a smooth curve.
For \(Y_1 = \text{ln} \frac{x}{2}\), you might choose values for \(x\) like 1, 2, 4, etc., and then calculate \(y\).
Graphing \(Y_2 = \text{ln} x - \text{ln} 2\) will produce the same graph because, given logarithmic properties, \(Y_2 = \text{ln} \frac{x}{2}\). Therefore, \(Y_1\) and \(Y_2\) are identical functions with identical graphs. Their graph is a smooth curve that increases slowly as \(x\) increases and passes through the point (1, -\text{ln}(2)).

Algebraic manipulation involves using mathematical techniques to simplify expressions or solve equations. For logarithmic functions, it includes applying logarithmic properties to rewrite and simplify expressions. In the exercise, we manipulated \(Y_2 = \text{ln} x - \text{ln} 2\) using:

• The quotient property: \( \text{ln} \frac{a}{b} = \text{ln} a - \text{ln} b \)

This simplification allows us to directly compare \(Y_2\) to \(Y_1\). Identifying and utilizing these properties can make complex logarithmic expressions more approachable. It helps with solving equations and comparing functions, as seen with \(Y_1\) and \(Y_2\) yielding the same function.