91Ó°ÊÓ

Understanding logarithmic functions is essential when solving logarithmic inequalities. A logarithmic function is the inverse of an exponential function and is expressed as \( y = \text{log}_b(x) \), where \( b \) is the base of the logarithm, \( x \) is the argument, and \( y \) is the logarithm of \( x \) to the base \( b \). In simple terms, it answers the question: 'To what power must we raise \( b \) to obtain \( x \)?'

For bases greater than 1, logarithmic functions increase as \( x \) increases. However, for bases between 0 and 1, such as \( \frac{1}{2} \) in our exercise, the function decreases with increasing \( x \), meaning that as the argument of the logarithm gets larger, the logarithm itself gets smaller. This behavior plays a crucial role in determining the way inequalities involving logarithms are solved, as it affects the direction of the inequality upon transformation.

When it comes to inequality solving, the process often involves isolating the variable of interest on one side to determine its range of possible values. The goal is to find all the values of the variable that make the inequality true.

For logarithmic inequalities like \( \text{log}_{\frac{1}{2}}(2x+3)>0 \), the process begins with understanding the behavior of the logarithmic function involved. Because logarithmic functions can behave differently according to their base, recognizing whether the base is greater or less than one is key. Then, we aim to remove the logarithm by using the properties of logs and exponents, often resulting in transforming the log inequality into a more familiar linear or polynomial inequality.

It's also important to remember that when multiplying or dividing by negative numbers, the direction of the inequality must be reversed. However, in our exercise since we're dealing with a base which is less than 1 and hence a decreasing function, the inequality direction remains the same when we applied the inverse operation. Always double-check your final answers to ensure they're within the domain of the logarithmic function, as logarithms are only defined for positive arguments.

Exponent rules, also known as the laws of exponents, are crucial for transforming and simplifying mathematical expressions involving powers. These rules provide the means to manipulate exponents in a variety of ways, ensuring that expressions and equations can be simplified and solved accurately.

Some of the fundamental exponent rules include:

• The product rule: \( a^m \times a^n = a^{m+n} \)
• The quotient rule: \( \frac{a^m}{a^n} = a^{m-n} \) when \( a eq 0 \)
• The power rule: \( (a^m)^n = a^{mn} \)
• The zero exponent rule: \( a^0 = 1 \) for any non-zero \( a \)
• The negative exponent rule: \( a^{-n} = \frac{1}{a^n} \) for any non-zero \( a \)

In the context of our exercise, we applied the power rule, converting the logarithmic statement into an equivalent exponential form. This allowed us to simplify the inequality \( (2x+3) > \frac{1}{2}^0 \) which, due to the zero exponent rule, simplifies further to \( (2x+3) > 1 \). A firm grasp of these rules is necessary when performing a variety of mathematical operations and they form a backbone for solving more complex problems involving exponents and logarithms.