91Ó°ÊÓ

Logarithms have some unique and helpful properties that can simplify solving equations. One key property is that if the natural logarithm (ln) of two numbers is equal, then the numbers themselves must be equal. In other words, for any numbers \(A\) and \(B\), if \(\ln(A) = \ln(B)\), then \(A = B\).

This property allows us to take logarithmic equations and convert them into simpler algebraic equations that are easier to solve. Let's see how this applies to the given exercise: \(\ln(x-3) = \ln(2x-9)\). Because the natural log of \(x-3\) equals the natural log of \(2x-9\), we can set the arguments equal, leading to \(x-3 = 2x-9\) which we can then solve algebraically.

Linear equations are equations of the first degree, which means they involve variables raised to the power of one. These equations are often in the form \(ax + b = 0\). In the given problem, the relationship between \(x-3\) and \(2x-9\) once the logarithmic property is applied gives us a linear equation.

Start by setting the two arguments (from the logarithms) equal to each other: \(x - 3 = 2x - 9\). To solve for \(x\), you need to isolate the variable on one side of the equation. Subtracting \(x\) from both sides, we get \(-3 = x - 9\). Then, by adding 9 to both sides, we find \(6 = x\). Linear equations like this one are straightforward and only require basic algebraic manipulations.

Verifying solutions is a crucial step in any mathematical problem to ensure the accuracy of your answer. After finding the solution \(x = 6\), we should substitute it back into the original equation to see if it holds true.

For our equation \(\ln(x-3) = \ln(2x-9)\), plug in \(x = 6\):

• \(\ln(6-3) = \ln(3)\)
• \(\ln(2(6)-9) = \ln(3)\)

Both sides simplify to \(\ln(3)\), confirming that the solution \(x = 6\) is correct. Verifying by checking the solution against the original equation ensures there were no mistakes made during the steps of solving.