91Ó°ÊÓ

Absolute value equations are a type of mathematical equation involving an absolute value expression. The absolute value of a number represents its distance from zero on a number line, without considering the direction. Hence, the absolute value of a number is always non-negative. For example, \(|x| = 3\) implies either \(x = 3\) or \(x = -3\). When solving absolute value equations, it's crucial to consider both the positive and negative scenarios that could yield the given absolute value.

• Identify the expression within the absolute value.
• Create separate equations for situations where the expression is non-negative and where it is negative.
• Solve each equation separately.

In our original exercise, we dealt with \(\left|2^x - 1\right|\). Here, solving involves considering two cases: \(2^x - 1 \geq 0\) and \(2^x - 1 < 0\). For these cases, handle the absolute value by stepping through the positive and negative versions of the expression to ensure all possibilities are covered.

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. These equations can have two solutions, one solution, or no real solutions depending on the discriminant \(b^2 - 4ac\).
Solving quadratic equations can be done using several methods:

• Factoring the polynomial if possible.
• Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
• Completing the square to transform the equation into an easier form.

In the exercise, we used factoring to solve quadratic equations that arose in both case analyses. For example, in Case 1, we dealt with \(y^2 - 3y - 4 = 0\). Factoring this equation, \(y = 4\) and \(-1\) were identified, but only \(y = 4\) was valid since \(y = 2^x\) must be positive.

Logarithms are the inverse operations of exponentials. They allow us to solve exponential equations by transforming them into a more manageable form. When you have an equation like \(b^x = y\), it can be rewritten using logarithms as \(x = \log_b(y)\).
The properties of logarithms include:

• Product rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\)
• Quotient rule: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)
• Power rule: \(\log_b(m^n) = n \cdot \log_b(m)\)

In our solution process, we arrived at a scenario where we needed to find \(x = \log_2(3)\), which came from solving the equation \(2^x = 3\). This example illustrates how logarithms can effectively deal with equations involving exponentials, converting them into a format that easily reveals the value of the variable of interest.