91Ó°ÊÓ

The power rule of logarithms is a fundamental property that helps simplify logarithmic expressions. This rule states that when you have a logarithm with an exponent, you can bring the exponent in front of the log. The formal rule is written as: \ \( \ln(a^b) = b \ln(a)\.\)
For example, consider \( \ln(x^2) \). Using the power rule, we can rewrite it as \( 2 \ln(x) \).
This property is valuable because it allows us to transform a complex logarithmic term into a simpler multiplication problem.
In the given exercise, we applied the power rule to \( \ln(x^2)\) and \( \ln(x^4)\), changing the original expression from \( \ln(x) - \ln(x^2) + \ln(x^4)\) to \( \ln(x) - 2 \ln(x) + 4 \ln(x)\). If you get comfortable with this rule, it will significantly streamline your work with logarithmic expressions.

After applying the power rule, the next step is to combine the logarithmic terms. When you have multiple logs with the same base, you can combine them by adding or subtracting the coefficients.
In our example, we started with \( \ln(x) - 2 \ln(x) + 4 \ln(x)\). Notice that each term has \( \ln(x)\) as its base.
To combine these, simply perform the arithmetic operations on the coefficients:
\( 1 - 2 + 4 = 3 \).
So, the expression now simplifies to \( 3 \ln(x)\).
This technique of combining like terms consolidates multiple log terms into a single, more manageable term. It is a useful skill when simplifying more complex equations.

The final step in the given exercise is to simplify the logarithmic expression.
After combining the terms, we have \( 3 \ln(x)\). This is the simplest form of the given expression because we’ve combined all logarithmic terms into one.
Simplifying expressions often involves:

• Applying relevant logarithmic rules and properties like the power and product rules;
• Performing basic arithmetic operations on coefficients;
• Ensuring the final form has no redundant elements.

By consistently applying these principles, you can tackle complex logarithmic problems with greater ease.
Remember, the goal is to make the expression as simple and clear as possible while preserving its value. This will help you solve equations and understand logarithmic relationships better.