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Solving logarithmic equations often involves applying the One-to-One Property of logarithms. This property is based on the idea that if the logarithms of two expressions with the same base are equal, then the expressions themselves are equal. As in our exercise, \(\log (2x+1) = \log 15\), we're implicitly working with the common logarithm base of 10.

By setting \(2x + 1\) equal to 15, we used the One-to-One Property to effectively 'remove' the logarithm and reduce the problem to a basic algebraic equation. It's crucial to recognize situations where this property applies; otherwise, solving these equations would require more complex algebraic manipulations or even reaching for graphing methods, which may not be as efficient.

The properties of logarithms are tools that help simplify complex logarithmic expressions and solve logarithmic equations. Key properties include the Product Rule, \(\log_b(mn) = \log_b(m) + \log_b(n)\), the Quotient Rule, \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\), and the Power Rule, \(\log_b(m^n) = n \log_b(m)\). In addition to these, the Change of Base formula and the One-to-One Property are essential when working with logarithms.

In the given exercise, we didn't need to apply these properties because the logarithmic expressions on both sides of the equation were already isolated. However, in more complex scenarios, these properties allow us to condense or expand logarithms to make equations easier to solve and to extract the variable of interest for further manipulation.

To isolate the variable means to manipulate the equation so that the variable is by itself on one side of the equation. This process is fundamental to solving any algebraic equation, including logarithmic equations.

When we subtracted 1 from both sides of the equation, \(2x + 1 = 15\) became \(2x = 14\), thus moving closer to isolating \(x\). Finally, by dividing by 2, we fully isolated \(x\), which yielded the solution \(x = 7\). It is often helpful to imagine the variable as the 'target' that needs to be freed from all other numbers and operations, so focusing on the reverse order of operations (undoing addition/subtraction followed by division/multiplication) will guide you toward successfully isolating the variable.