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Logarithmic functions are fascinating and a crucial part of mathematics. They involve understanding how a logarithm works. A logarithm answers the question: "To what exponent must a base be raised to produce a certain number?" The natural logarithm, denoted by \( \ln \), is a specific type of logarithm with a base of \( e \), where \( e \approx 2.71828 \).

Logarithmic functions are only defined for positive inputs. This is important for determining the domain of such functions. For example, \( \ln (x) \) is only defined when \( x > 0 \). When tackling problems involving logarithmic functions, consider:

• Ensuring the input to the logarithmic function is positive.
• Setting up inequalities to determine this domain.

Understanding these concepts will help you solve problems involving logarithmic functions, which appear in both pure and applied mathematics.

Inequalities are mathematical statements that describe the relationship between two values. These can be represented by signs such as \( >, <, \geq, \text{and} \leq \). In the context of determining the domain of a function like \( f(x)=\ln (6-x) \), setting up an inequality helps find the range of permissible values for \( x \).

When solving inequalities, remember:

• To flip the inequality sign when multiplying or dividing both sides by a negative number, making sure the solution remains valid.
• Step-by-step simplification is crucial. For instance, converting \( 6-x > 0 \) to \( x < 6 \) shows which values make \( 6-x \) positive, essential for defining the function.

Inequalities help us express ranges of numbers, crucial when working with domains, ensuring clarity in solutions.

Real numbers are a broad category of numbers used to express continuous values. They include all the rational and irrational numbers on the number line. Rational numbers like integers and fractions, and irrational numbers such as \( \pi \) and \( \sqrt{2} \), all fall under real numbers.

When determining a function's domain, as in the case of \( f(x)=\ln (6-x) \), you might restrict the domain to certain real numbers. For instance, the domain here is all real numbers less than 6.

Real numbers encompass:

• Positive and negative numbers.
• Whole numbers and decimals.
• Numbers that can be plotted on a number line.

Understanding real numbers allows us to accurately interpret a function's domain and helps ensure a function is meaningfully defined within its applicable range.