What is the solution to this equation?
$\ln (x+6)-\ln (2 x-1)=0$
The goal is to solve the equation \(\ln(x+6) - \ln(2x-1) = 0\). First, we identify the domain of the expression. For the logarithmic functions \(\ln(x+6)\) and \(\ln(2x-1)\) to be defined, their arguments must be strictly positive. Thus, we require \(x+6 > 0\) and \(2x-1 > 0\). Solving these inequalities, we find \(x > -6\) and \(x > 0.5\). The intersection of these conditions implies that the solution must satisfy \(x > 0.5\).
We rewrite the original equation \(\ln(x+6) - \ln(2x-1) = 0\) by adding \(\ln(2x-1)\) to both sides, which gives us \(\ln(x+6) = \ln(2x-1)\). Since the natural logarithm function is one-to-one, \(\ln(A) = \ln(B)\) implies that \(A = B\). Therefore, we set the arguments equal to each other: \(x+6 = 2x-1\).
Now we solve the linear equation \(x + 6 = 2x - 1\) for \(x\). Subtracting \(x\) from both sides gives \(6 = x - 1\), and adding 1 to both sides yields \(x = 7\). We check this value against our domain constraint, \(x > 0.5\). Since \(7 > 0.5\), the value \(x = 7\) is a valid solution. Comparing this to the given options (\(x=-5, 7, -7, 5\)), we see that \(x=7\) is the correct choice.
Logarithms are widely used in fields like chemistry and finance. For instance, the pH scale used to measure the acidity of solutions is based on logarithmic relationships, defined as \(pH = -\log_{10}[H^+]\), where \([H^+]\) is the concentration of hydrogen ions. Understanding how to solve logarithmic equations allows scientists to determine reactant concentrations or decay rates, ensuring precise results in experimental and real-world applications. This same algebraic logic helps in calculating compound interest and modeling exponential growth in population dynamics.