Solve the equations by finding the exact solution.
ln\(x\) - ln\(9\) = 3
To solve the equation ln(x) - ln(9) = 3, we can use the properties of logarithms and algebraic manipulation.
Step 1: Combine the logarithms on the left side of the equation using the quotient rule of logarithms. The quotient rule states that ln(a) - ln(b) = ln(a / b).
ln(x / 9) = 3
Step 2: Convert the equation from logarithmic form to exponential form. The exponential form of ln(y) = z is y = e^z. In this case, we have ln(x / 9) = 3, so we can write x / 9 = e^3.
x / 9 = e^3
Step 3: Multiply both sides of the equation by 9 to isolate x.
x = 9 * e^3
Step 4: Evaluate the right side of the equation.
Using a calculator or mathematical software, approximate e^3 to get:
x ≈ 9 * 20.08553692
x ≈ 180.7698323
Therefore, the exact solution to the equation ln(x) - ln(9) = 3 is x ≈ 180.7698323.