What is the solution of $\ln (x+6)-\ln 9=2$ ? Round your answer to the nearest hundredth.
We are given the logarithmic equation \(\ln(x+6) - \ln 9 = 2\). Our primary goal is to isolate the variable \(x\). First, we note the domain constraint: the argument of the logarithm must be positive, so \(x+6 > 0\), which implies \(x > -6\). To proceed, we use the quotient rule for logarithms, which states \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\). This allows us to combine the two logarithmic terms into one:
\[\ln\left(\frac{x+6}{9}\right) = 2\]
Next, we convert the equation from its logarithmic form to its equivalent exponential form. The rule \(\ln(y) = k\) is equivalent to \(y = e^k\). Applying this to our equation:
\[\frac{x+6}{9} = e^2\]
To isolate \(x\), we first multiply both sides by 9:
\[x+6 = 9e^2\]
Then, we subtract 6 from both sides to obtain:
\[x = 9e^2 - 6\]
Now, we calculate the numerical value of \(x\) using the approximation \(e \approx 2.71828\).
Calculating \(e^2 \approx 7.389056\).
Then, \(x = 9 \times 7.389056 - 6 \approx 66.5015 - 6 = 60.5015\).
Rounding this value to the nearest hundredth gives \(x = 60.50\). Checking this against the given options, we see that it corresponds to option a.
Logarithmic equations are used in various fields such as finance, acoustics, and chemistry. For example, the pH level of a solution is determined by the negative logarithm of the hydrogen ion concentration. In finance, logarithms help solve for time periods in compound interest problems where the future value, present value, and interest rate are known. Understanding how to manipulate these equations allows for precise calculations in these real-world applications.