Question

\(\left.\ln n^{n}\right)=\ldots \ldots\)

Short answer

The simplification of \( \ln n^n \) is \( n \ln n \).

Step-by-step solution

Identify the Expression

The given expression is \( \ln n^n \). We need to simplify this expression using logarithmic properties.

Apply the Power Rule for Logarithms

Recall that the power rule for logarithms states that \( \ln a^b = b \cdot \ln a \). In this case, \( a = n \) and \( b = n \).

Simplify the Result

Applying the power rule, we have \( \ln n^n = n \cdot \ln n \). There are no further simplifications possible.

Key concepts

Power Rule for Logarithms

When dealing with logarithms, the power rule is extremely useful for simplification. This rule states that if you have a logarithm of a number raised to a power, such as \( \ln a^b \), it can be simplified to \( b \cdot \ln a \). This is because the logarithm of an exponent allows us to move the exponent to the front as a multiplier. This property comes from the laws of exponents and how they relate to logarithms. By simplifying like this, calculations become more manageable, especially for larger numbers.

• Identify the base \( a \) and the exponent \( b \).
• Use the formula: \( \ln a^b = b \cdot \ln a \).

By using the power rule, we can simplify complex expressions, making them easier to work with or further manipulate if needed.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is all about making the expression easier to handle while retaining its value. Once we've applied the power rule for logarithms, the next step is to simplify wherever possible. Let's say you have already transformed \( \ln n^n \) to \( n \cdot \ln n \) using the power rule. Now, the expression appears in its simplest form, but always double-check for additional simplification.

Keep an eye out for common factors or additional rules that may apply, such as:

• Product Rule: \( \ln( a \cdot b ) = \ln a + \ln b \)
• Quotient Rule: \( \ln( \frac{a}{b} ) = \ln a - \ln b \)

Being aware of different rules helps transform the expression into a simpler and clearer form.

Natural Logarithm

The natural logarithm, denoted \( \ln(x) \), is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.718. It's frequently used in mathematics due to its natural occurrence in processes describing growth, decay, and in various natural phenomena. \( \ln(x) \) is especially powerful because it simplifies expressions involving exponentials and is the inverse function of the exponential function \( e^x \).
Using natural logarithms, complex expressions like those involving exponential growth can be resolved into simpler forms. This is why they are central in calculus, financial modeling, and many scientific disciplines.

• The natural logarithm of \( x \) is the power to which \( e \) must be raised to equal \( x \).
• \( \ln(1) = 0 \) because \( e^0 = 1 \).

Understanding \( \ln \) helps in simplifying expressions and solving equations that would otherwise be complicated.