Problem 51 - 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. $$\ln x=2$$

Step-by-Step Solution

Answer

The exact answer is \(x = e^2\), and the decimal approximation for the solution is 7.39.

Step 1: Identify the Logarithm Base

The given equation is \(\ln x = 2\). The logarithm here is a natural logarithm, denoted by \(\ln\), which implies that the base of the logarithm is the number \(e\), where \(e\) is approximately equal to 2.71828.

Step 2: Rewrite in Exponential Form

Rewrite the logarithmic equation in exponential form. According to the definition of logarithms, we know that \(\ln a = b\) is equivalent to \(e^b = a\). So, the given equation \(\ln x = 2\) can be rewritten in exponential form as \(e^2 = x\).

Step 3: Calculate the Value of \(x\)

To find the value of \(x\), calculate \(e^2\). It gives the exact value of \(x = e^2\).

Step 4: Decimal Approximation

For the decimal approximation, use a calculator to calculate \(e^2\) which should be correct to two decimal places. So the approximation of \(e^2\) up to two decimal places is 7.39.