To solve the equation [tex]\(\ln (x + 6) - \ln 9 = 2\)[/tex], we need to isolate [tex]\(x\)[/tex]. Here's the detailed step-by-step solution:
1. Understand the properties of logarithms:
[tex]\[\ln a - \ln b = \ln \left(\frac{a}{b}\right)\][/tex]
Applying this property to our equation gives:
[tex]\[\ln (x + 6) - \ln 9 = \ln \left(\frac{x + 6}{9}\right) = 2\][/tex]
2. Exponentiate both sides to eliminate the natural logarithm [tex]\(\ln\)[/tex]:
[tex]\[e^{\ln \left(\frac{x + 6}{9}\right)} = e^2\][/tex]
Since [tex]\(e^{\ln k} = k\)[/tex], we have:
[tex]\[\frac{x + 6}{9} = e^2\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[\frac{x + 6}{9} = e^2\][/tex]
Multiply both sides by 9:
[tex]\[x + 6 = 9e^2\][/tex]
Subtract 6 from both sides:
[tex]\[x = 9e^2 - 6\][/tex]
4. Calculate [tex]\(e^2\)[/tex]:
[tex]\(e \approx 2.71828\)[/tex], so:
[tex]\[e^2 \approx (2.71828)^2 \approx 7.3891\][/tex]
5. Multiply [tex]\(9 \cdot e^2\)[/tex]:
[tex]\[9 \cdot 7.3891 \approx 66.5019\][/tex]
6. Subtract 6:
[tex]\[x = 66.5019 - 6 \approx 60.5019\][/tex]
Rounding to the nearest hundredth:
[tex]\[x \approx 60.50\][/tex]
Therefore, the solution to the equation [tex]\(\ln (x + 6) - \ln 9 = 2\)[/tex] rounded to the nearest hundredth is:
[tex]\[ \boxed{60.50} \][/tex]
