Problem 65 - 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. $$\ln \sqrt{x+3}=1$$

Step-by-Step Solution

Answer

The exact answer to the equation is \(x = e^2 - 3\) and the decimal approximation for this solution, rounded to two decimal places, will need a calculator depending upon the precision of \(e\). The domain of the answer is valid.

Step 1: Rewrite the logarithmic equation

The logarithmic equation can be rewritten as an equivalent exponential equation. In this case, \(\ln \sqrt{x+3}=1\) is equivalent to \(e^1 = \sqrt{x+3}\) due to the nature of natural logs being the logarithm to the base \(e\). Hence the equation results in \(e = \sqrt{x+3}\).

Step 2: Square both sides

To eliminate the square root, square both sides of the equation. This gives \((e)^2 = (\sqrt{x+3})^2\), which simplifies to \(e^2 = x+3\).

Step 3: Solve for \(x\)

To solve for \(x\), subtract 3 from both sides of the equation. This gives \(x = e^2 -3\). Now, compute the actual decimal value for the right hand side expression to get the numerical value for \(x\).

Step 4: Verify domain

Lastly, we must validate that the solution of \(x\) lies within the domain of the original logarithmic expression. It does, because \(x\) is less than -3, which is the boundary for the original log's domain. Therefore, the solution is valid.