What is the solution to this equation?
$\ln (x+6)-\ln (2 x-1)=0$
To solve the equation \(\ln(x+6) - \ln(2x-1) = 0\), we first identify the domain of the expression. The logarithmic function \(\ln(u)\) is defined only for \(u > 0\). Therefore, we must satisfy the following conditions:
\[x + 6 > 0 \implies x > -6\]
\[2x - 1 > 0 \implies x > \frac{1}{2}\]
Combining these conditions, we conclude that the variable \(x\) must be greater than \(\frac{1}{2}\). Any valid solution must fall within this domain.
We can rewrite the original equation using the property of logarithms which states that \(\ln(A) - \ln(B) = \ln(\frac{A}{B})\):
\[\ln\left(\frac{x+6}{2x-1}\right) = 0\]
By definition, \(\ln(u) = 0\) if and only if \(u = 1\). Thus, we set the argument of the logarithm to 1:
\[\frac{x+6}{2x-1} = 1\]
Now, we solve the rational equation for \(x\). Multiplying both sides by the denominator \(2x - 1\):
\[x + 6 = 2x - 1\]
Subtracting \(x\) from both sides gives:
\[6 = x - 1\]
Adding 1 to both sides gives:
\[x = 7\]
We must check if this value is within the established domain \(x > \frac{1}{2}\). Since \(7 > \frac{1}{2}\), the solution is valid.
We have found that \(x = 7\). Comparing this to the given choices, we see that \(x=7\) is one of the options. Therefore, the correct solution to the equation is \(7\). The final answer is \(\boxed{7}\).
Logarithmic equations are used in various fields like chemistry to calculate the pH of solutions, where \(\text{pH} = -\log[H^+]\), or in finance to determine the time required for an investment to grow to a certain amount using compound interest formulas. By solving equations like the one provided, we learn to navigate domain restrictions and logarithmic identities, which are fundamental in modeling real-world exponential growth and decay processes.