Problem 62 - 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. $$7+3 \ln x=6$$

Step-by-Step Solution

Answer

The exact solution to the equation \(7+3 \ln x = 6\) is \(x = e^{(-1 / 3)}\), which approximates to \(x = 0.73\)

Step 1: Isolate the logarithmic term

To isolate the logarithmic term, the constant 7 is subtracted from both sides of the equation, leading to \(3 \ln x = 6 - 7\). Simplifying the right-hand side gives \(3 \ln x = -1\).

Step 2: Divide by coefficient

The coefficient or the numerical factor of the logarithmic term is 3. This is divided on both sides of the equation to get \(\ln x = -1 / 3\).

Step 3: Transform the Logarithmic Equation to an Exponential Equation.

The logarithmic equation \(\ln x = -1 / 3\) can be transformed into an equivalent exponential equation. The base of the natural logarithm is 'e', thus \(x = e^{(-1 / 3)}\).

Step 4: Check the solution

The obtained solution, \(x = e^{(-1 / 3)}\), must be in the domain of the original logarithmic function ln(x), which is \(x > 0\). Since \(e^{(-1 / 3)} > 0\), it is within the domain and therefore a valid solution.

Step 5: Obtain a decimal approximation

The exact solution \(x = e^{(-1 / 3)}\) can be approximated to two decimal places using a calculator to get \(x = 0.73\).