91Ó°ÊÓ

Understanding the properties of logarithms is essential when it comes to solving logarithmic equations. Logarithms, in essence, are exponents. The equation \(\log_2(3x-4) = \log_2(5)\) from the exercise relies on the property that if two logarithms with the same base are equal, their arguments must also be equal. This property reflects the one-to-one nature of logarithm functions. Simplifying logarithmic expressions frequently leads to solving the argument of the logarithms, as seen in the featured equation where we equate \(3x - 4\) to 5 directly due to the equality of their logarithmic forms.

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. This involves a variety of techniques such as distributing, combining like terms, adding, subtracting, multiplying, and dividing both sides of equations to isolate the unknown variable. In our example, once we ascertain that \(3x - 4 = 5\), we proceed by performing simple algebraic operations: adding 4 to both sides to eliminate the constant term on the left side and isolating the term containing the variable. These steps transform the equation into a simpler form, \(3x = 9\), reducing the problem to a basic algebraic equation solvable with elementary skills.

Isolating the variable is a fundamental goal in solving any algebraic equation. It means rearranging the equation so that the variable we are solving for is on one side of the equation, and everything else is on the other side. In the given example, after using logarithm properties and algebraic manipulation, we eventually position \(x\) by itself on one side through a sequence of operations. This process begins by adding or subtracting terms to remove constants from the side of the variable and is followed by multiplying or dividing to remove coefficients. In the final step for our equation, we divide both sides by 3, yielding \(x = 3\), thereby isolating \(x\). Understanding how to effectively isolate variables is critical, as it is the last step in solving for an unknown in most algebraic equations.