Problem 55 - Introductory and Intermediate Algebra for College Students 4th

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. $$\ln x=2$$

Step-by-Step Solution

Answer

The solution to the equation \(\ln x=2\) is \(x \approx 7.39\).

Step 1: Conversion to Exponential Form

The equation \(\ln x=2\) is in logarithmic form. We'll convert this into its equivalent exponential form. The base of natural logarithm \(\ln\) is \(e\), so the equivalent exponential form would be \(e^2=x\).

Step 2: Calculating the Value of x

Now, calculate the exponentiation of the base \(e\) raised to \(2\) which equals to \(x\). This is done using a scientific calculator. Using a calculator, we get \(e^2 \approx 7.39\).

Step 3: Checking the Validity of the Solution

The value \(x = 7.39\) is a positive real number which satisfies the condition for the real logarithms \(x > 0\). Hence, this solution is valid.