Problem 56 - 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. $$\ln x=3$$

Step-by-Step Solution

Answer

The exact solution of the equation \(\ln x = 3\) is \(x = e^3\). The decimal approximation of the exact solution \(x = e^3\) is \(x = 20.08\).

Step 1: Understand the equation

Firstly, understand the given equation \(\ln x = 3\). This equation represents the natural logarithm of \(x\) with base \(e\) and its value is given as 3.

Step 2: Convert the equation

Convert the logarithmic equation into an exponential equation. By definition of logarithms, the equation \(\ln x = 3\) is equivalent to the exponential equation \(e^3 =x\).

Step 3: Calculate the value of x

Solve the exponential equation to get the value of \(x\). Recall that 'e' is a mathematical constant which is approximately equal to 2.71828. Therefore, \(x\) is equal to \(e^3 = 2.71828^3\).

Step 4: Approximate the answer

Finally, approximate the value of \(x\) to two decimal places using a calculator, which is around 20.08.