91Ó°ÊÓ

Factoring equations is a fundamental technique in algebra that involves expressing a polynomial or mathematical expression as a product of its factors. When given an equation like \( y^3 = 3y \), our goal is to simplify and solve it by finding the values for \( y \). In this case, we notice that each term in the equation has a common factor. By factoring, we rewrite this as:

• \( y(y^2 - 3) = 0 \)

This equation indicates that either \( y = 0 \) or \( y^2 - 3 = 0 \). The process of factoring helps break down the equation into simpler parts, making it easier to find solutions. It's important to check for and consider trivial solutions and decompositions to ensure that all possible solutions are found. This method is widely applied in solving quadratic, cubic, or higher degree equations.

Once an equation is factored, the next step is to solve for the unknowns involved. When we have \( y(y^2 - 3) = 0 \), it implies two potential solutions:

• \( y = 0 \) leads to \( \log x = 0 \), and since \( 10^0 = 1 \), we find \( x = 1 \).
• \( y^2 - 3 = 0 \) can be solved by adding 3 to both sides and taking the square root, yielding \( y = \pm\sqrt{3} \).

By solving these equations, we determine the corresponding values of \( x \). It's crucial to isolate the unknown variable step by step and reflect each manipulation clearly to avoid mistakes. Understanding the relationship between the original variables and their logarithmic expressions is key in deriving the correct solutions.

Logarithms are an essential concept in mathematics used to represent and solve exponential equations. The core property that we use here is that \( \log_{10}(x) = y \) means \( x = 10^y \), which helps us convert between logarithmic and exponential forms.

• For \( y = 0 \), we know \( x = 10^0 = 1 \).
• For \( y = \sqrt{3} \), we find \( x = 10^{\sqrt{3}} \).
• For \( y = -\sqrt{3} \), it translates to \( x = 10^{-\sqrt{3}} \).

Understanding and applying these properties allows us to evaluate logarithmic equations effectively. Logarithms turn multiplication into addition, division into subtraction, and exponentiation into a simple multiplication, which simplifies solving even complex equations. Always remember to verify that the found solutions satisfy the initial conditions to ensure they are valid.