Question

what is the solution of \\( \ln ( x + 6 ) - \ln 9 = 2 \\)? round your answer to the nearest hundredth. \\( x = 5 3. 5 0 \\) \\( x = 8 0. 5 0 \\) \\( x = 8 6. 5 0 \\) \\( x = 7 2. 5 0 \\)

Explanation:

Step1: Apply logarithm property

Use the property \(\ln a-\ln b=\ln\frac{a}{b}\). So, \(\ln(x + 6)-\ln9=\ln\frac{x + 6}{9}\). The equation becomes \(\ln\frac{x+6}{9}=2\).

Step2: Convert from logarithmic to exponential form

Since \(y = \ln z\) is equivalent to \(z=e^{y}\) (where \(e\approx2.71828\)), we have \(\frac{x + 6}{9}=e^{2}\).

Step3: Solve for \(x\)

First, multiply both sides by \(9\): \(x+6 = 9e^{2}\). Then, \(x=9e^{2}-6\). Substitute \(e^{2}\approx7.389\), so \(x=9\times7.389-6\).
Calculate \(9\times7.389 = 66.501\), then \(x=66.501-6=60.501\approx60.50\)

Answer:

\(x = 60.50\)