91Ó°ÊÓ

Logarithmic functions are special and widely used in mathematics, especially in calculations involving growth rates, regression analysis, and even in computer science logics. A logarithmic function is in the form of usually either the natural logarithm, denoted as \(\ln(x)\), or the common logarithm, \(\log_{10}(x)\). These are inverse functions of exponential functions, meaning that they allow us to express powers in terms of multipliers.

• The key point to remember about logarithmic functions is that they require their arguments to be greater than zero.
• This is because a log function asks "what power do I need to raise the base to in order to get this number?" and for numbers less than or equal to zero, there is no valid answer.

In more straightforward terms, when you encounter a function like \(y = \ln(u)\) or \(y = \log_{10}(u)\), the expression inside the log, represented here as \(u\), must always fulfill the condition \(u > 0\).
Knowing this, when tasked with finding the domain of a logarithmic function, begin by determining where this inequality is satisfied by the expression inside the log. This analysis helps us confidently establish safe values for \(x\) that make the logarithmic function valid.

Solving inequalities is a crucial part of determining domains for logarithmic functions. When faced with inequalities, our task is to find all possible values for which the inequality holds true. This can include complex expressions but breaks down into logical steps for simplicity.

• Begin by rearranging the inequality if necessary for clarity. Place all terms on one side of the inequality. This puts the expression into a standard form such as \(f(x) > 0\).
• For quadratic inequalities, like \(-(x+1)(x+2) < 0\), the process starts with finding the critical values or roots, which happen when the expression equals zero.
• These roots will split the number line into intervals. Test each interval to see if the inequality holds.

For example, with function (a) \(2 - x - x^2 > 0\), the expression can be rearranged and factored into \((x - 1)(x + 2) < 0\). We find that the roots, or critical points, are \(x = 1\) and \(x = -2\).
To determine where the inequality is true, test values from each interval created by these points. You'll establish regions like \((-\infty, -2)\), \((-2, 1)\), and \((1, \infty)\), and check which ones satisfy the inequality.

Determining the domain of a function involves identifying all possible inputs (or \(x\)-values) that will produce real, valid outputs. For logarithmic functions, this determination means solving for where the argument inside the log can be greater than zero.

• Let's recall, function domains are crucial in real-world applications because they define where the function can be appropriately applied.
• It provides the range where all transformation and function manipulations will maintain their integrity.

For function (a) \(y=\ln(2-x-x^2)\), the domain requires solving \(2-x-x^2>0\) to find the valid \(x\)-values. Using factorization and test intervals, we find that the domain is \((-2, 1)\).
In function (b) \(y=\log_{10}\left(\frac{2x+3}{x-5}\right)\), prioritize solving \( \frac{2x+3}{x-5} > 0\). Find that the function is undefined at \(x = 5\) and has a zero at \(x = -\frac{3}{2}\). Thus, the valid domain is \((-\infty, -\frac{3}{2}) \cup (5, \infty)\).
Through these examples, you see that establishing a domain is visually about finding these critical intervals and understanding where the function "works" as intended.