Problem 52 - College Algebra Essentials

Question

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln x=3$$

Step-by-Step Solution

Answer

Solving the equation \(\ln x = 3\) gives the value of \(x = e^3\). On calculating, \(x \approx 20.09\).

Step 1: Convert the equation to exponential form

Using the logarithm rules and converting the logarithmic equation \(\ln x = 3\) to exponential form. It becomes \(e^3 = x\). (\(e\) is the base for natural logarithms)

Step 2: Solve for x

Now we can solve for x by calculating the value of \(e^3\). This computation can be achieved with the help of a scientific calculator.

Step 3: Verify the solution

Check if the solution for 'x' fits in the domain of the logarithmic function. Since natural logarithm is defined for all positive real numbers, the solution need to be greater than zero to be real and valid.