Problem 64 - 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. $$7+3 \ln x=6$$

Step-by-Step Solution

Answer

The exact solution is \(x=e^{-1/3}\) and the approximate solution is \(x \approx 0.71\)

Step 1: Isolate the Logarithmic Expression

Firstly, isolate the natural logarithm on one side of the equation by subtracting 7 from both sides: \[3 \ln x=6-7\] which simplifies to: \[3 \ln x=-1\]

Step 2: Make the Coefficient of ln x Equal to One

Next, to remove the 3 that is being multiplied with the logarithm, divide both sides of the equation by 3: \[\ln x=\frac{-1}{3}\]

Step 3: Express the Equation in Exponential Form

In order to get the variable out of the logarithmic function, we will need to convert this information into an exponential equation. Therefore, converting the equation above: \[x=e^{-1/3}\]

Step 4: Verify the Domain

The last step would be to verify if the calculated \(x\) value is within the domain of original logarithmic expressions. In this specific problem, it is given by \(\ln x\), and its domain is \(x > 0\). Since \(e^{-1/3} > 0\), the calculated answer is within the domain.

Step 5: Decimal Approximation

For decimal approximation, calculate the numerical value of \(x\) with a calculator up to two decimal places. The approximate value should be \(x \approx 0.71\)