91Ó°ÊÓ

Logarithmic functions are the inverse of exponential functions. If you understand that, you're already on the right path! The logarithm of a positive number in base 10 is the power to which 10 must be raised to obtain that number. In simpler terms, if \( b^y = x \, then \ \log_b (x) = y \). Let's use a familiar base, 10, for really grasping this:

• \ \log_{10} (100) = 2 \ because 10 squared (or \(10^2\)) is 100.

Logarithmic functions such as \(f(x) = \ln(x)\) specifically use the natural base \(e\). Here \(e \approx 2.718\), another constant often used in calculus. Remember: A logarithm is only defined for positive numbers. That means the input (the value for which you try to find the logarithm) must be greater than zero.
This crucial rule comes into play with the given exercise where you were asked to find out what makes \(4 - x^2\) greater than zero. This ensures that the function \(\ln(4 - x^2)\) will be valid and makes sense.

The domain of a function is basically all the possible input values \(x\) that allow the function to run without any hitches. Think of the domain like an all-access backstage pass to a concert—for those values of \(x\), the show must go on. This involves ensuring that inputs meet all the basic requirements of the function.

• For rational functions, the denominator cannot be zero.
• For square roots, the expression under the root must be non-negative.
• For logarithms, the argument must be positive.

Here, since you are dealing with a natural logarithm \(\ln(4-x^2)\), you need \(4-x^2 > 0\) to keep \(\ln\) happy.
In solving that inequality, you'd find that \(x\) lies within the open interval \(( -2, 2 )\). This interval defines the domain of \(f(x)\), meaning \(f(x)\) can only be evaluated for \(x\) values falling strictly between -2 and 2.

Inequalities allow us to express the range of values for which certain conditions of a function hold true. They are like the rules of the game, letting you know where you can play freely with expressions. When working with inequalities in calculus, the process usually involves:

• Isolating the variable.
• Performing operations like addition, subtraction, multiplication, or division (note: be cautious when multiplying or dividing by a negative number as it flips the inequality sign).
• Using algebraic manipulation to pinpoint intervals where the inequality holds true.

In our exercise, solving the inequality \(4-x^2 > 0\) involved rearranging terms to find \(x^2 < 4\). Taking square roots gives us two solutions: \(-2 < x < 2\). This interval specifies that the original condition, the positivity of \(4-x^2\), is met, thus ensuring the continuity of \(f(x)\). Remember: The inequality is strict (using \(>\) not \(\geq\)), so the endpoints are not included. This critical understanding of inequalities lets us carve out the domain for the function effectively.