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Logarithmic identities are powerful tools in mathematics that help simplify complex logarithmic expressions. To start, understand that logarithms and exponents are inverse operations, much like addition and subtraction or multiplication and division.
This means they can cancel each other out under specific circumstances. For instance, using the identity \( \,a \ln b = \ln b^a\, \), you transform multiplication into an exponentiation within the logarithm.
In the context of our exercise, this identity shows up when \(2 \ln x\) is rewritten as \(\ln x^2\). This conversion makes it easier to combine terms in subsequent steps.
To recognize when to use this identity, look for exponents or coefficients applied to logarithms in the form \(\ln\,b\).

Exponential functions feature constant rates of growth or decay, represented as a number raised to a variable power. These functions are key in mathematical modeling, from population dynamics to radioactive decay.
In our problem, the base of the natural logarithm, denoted \(e\), plays a central role. The expression \(e^{\ln a}\) can be simplified because the operations of taking a natural log and raising \(e\) to that power cancel each other out, essentially "undoing" one another.
This is a crucial simplification step: if you see \(e^{\ln (something)}\), know that it reduces to just \((something)\).

• This makes calculations quicker and easier.
• It also underlines the relationship between logarithms and exponents as inverse operations.

The properties of logarithms provide multiple ways to manipulate and simplify logarithmic expressions. In the context of our problem, a key property comes into play:

• \( \ln a + \ln b = \ln (ab)\)

This property allows you to add separate logarithms into a single expression, reflecting a multiplication of their inner terms.
In the exercise, after transforming \(2 \ln x\) into \(\ln x^2\), we proceed to combine \(\ln 3\) and \(\ln x^2\) using our property into \(\ln (3 x^2)\).
Using this property efficiently requires recognizing parts of an expression that can be consolidated, reducing the expression's complexity steps ahead.
Being familiar with this and other logarithmic properties allows mathematicians and students alike to simplify expressions with greater ease.