91Ó°ÊÓ

Solving equations involves finding the value of the variable that makes the equation true. In our original exercise, the equation is presented in logarithmic form: \( \log(3x) = \log(9) \). When faced with such equations, the understanding of the properties of logarithms is essential.

Since the logs have the same base, we can use the property that if the logarithms are equal, their arguments must be equal as well. This means the equation can be rewritten without logs as \( 3x = 9 \). Now, this simplifies our task to solving a basic linear equation, which is straightforward by dividing both sides by the coefficient of \( x \), giving us \( x = 3 \). This is a simple explanation of how solving logarithmic equations can often be reduced to more familiar types of equations like linear ones.

The properties of logarithms are powerful tools for solving logarithmic equations. Some of the key properties include:

• If \( \log_{b}(A) = \log_{b}(B) \), then \( A = B \). This property allows us to equate the arguments of the logarithms when their logarithms are equal.
• The product rule: \( \log_{b}(A \cdot B) = \log_{b}(A) + \log_{b}(B) \).
• The quotient rule: \( \log_{b}\left(\frac{A}{B}\right) = \log_{b}(A) - \log_{b}(B) \).
• The power rule: \( \log_{b}(A^n) = n \cdot \log_{b}(A) \).

In the original exercise, we utilize the property that allows us to equate arguments when the logarithmic expressions on both sides are equal. This converts the problem into an algebraic equation that is much easier to solve.

Algebraic manipulation is a fundamental skill in solving equations, including logarithmic ones. After applying logarithmic properties, we often transform the equation into a form that can be easily solved using algebra.

In the exercise \( 3x = 9 \), we used basic algebraic manipulation by isolating \( x \). Here are the steps:

• Divide both sides of the equation by 3, because we want \( x \) alone on one side. This operation gives us \( x = \frac{9}{3} \).
• Simplify to find \( x = 3 \).

Algebraic manipulation involves operations that rearrange and simplify equations, such as addition, subtraction, multiplication, and division. Mastering these skills is crucial for efficiently solving both simple and complex equations.