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Logarithmic properties are essential in simplifying and solving logarithmic expressions. These properties help us break down complex logarithms into simpler components.
One key property is the **product rule**, which states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, this is written as: \[ \text{ln}(xy) = \text{ln}(x) + \text{ln}(y) \] This means if you have the logarithm of two numbers multiplied together, you can split it into the sum of two simpler logarithms.
Another useful property is the **quotient rule**. It states that the logarithm of a fraction is the difference of the logarithms of the numerator and the denominator:\[ \text{ln}\bigg(\frac{x}{y}\bigg) = \text{ln}(x) - \text{ln}(y) \] This rule is incredibly useful when dealing with fractions, as seen in the given exercise.
Lastly, the **power rule** states that the logarithm of a number raised to an exponent is the exponent times the logarithm of the base:\[ \text{ln}(x^y) = y \times \text{ln}(x) \] These properties not only simplify computations but also help in solving logarithmic equations.

The natural logarithm, often written as \(\text{ln}\), is a special type of logarithm where the base is the constant **e** (approximately 2.71828). The natural logarithm is commonly used in various fields, especially in science and engineering.
One of the most notable aspects of the natural logarithm is how it simplifies the process of dealing with exponential growth and decay problems. Because its base is **e**, it has convenient properties for differentiation and integration in calculus.
In the context of the given exercise, we are dealing with natural logarithms. For example, we know that \(\text{ln}(2) = a\) and \(\text{ln}(3) = b\). Using the properties of logarithms, we can transform and simplify expressions involving the natural logarithm. This allows us to write complex logarithmic values in terms of known simpler values.

Working with logarithms of fractions can be simplified using the quotient rule of logarithms. In the exercise, we need to find \(\text{ln}(1.5)\), but first, we expressed 1.5 as the fraction \(\frac{3}{2}\).
This transformation is crucial because it allows us to apply the quotient rule:
\[ \text{ln}\bigg(\frac{3}{2}\bigg) = \text{ln}(3) - \text{ln}(2) \] By substituting the given values \(\text{ln}(2) = a\) and \(\text{ln}(3) = b\), we can now write:\[ \text{ln}\bigg(\frac{3}{2}\bigg) = b - a \] This simplification shows the power of logarithm properties in breaking down and understanding fractions in terms of logarithms.
These transformations make it easier to handle logarithmic equations and expressions, enhancing our ability to solve problems involving logarithms.