Problem 64 - Algebra and Trigonometry

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 solution to the given logarithmic equation is approximately \(x \approx 0.717\)

Step 1: Rearrange the equation

We start by trying isolating the logarithmic term. To do this, we take 7 from both sides of the equation to get: \[ 3 \ln x = 6 - 7 = -1 \]

Step 2: Convert to exponential form

Next, still isolating \(x\), we divide both sides by 3, resulting in: \[ \ln x = -\frac{1}{3} \]. Now we can convert the logarithmic equation into an exponential form. The equation gets converted to: \[ x = e^{(-\frac{1}{3})} \]

Step 3: Compute value using calculator

Finally, we calculate the value of \(x\), letting \(e\) to be approximately 2.71828, to find the decimal representation. Using a calculator for this calculation, we get: \[ x \approx 0.717 \]

Key Concepts

Domain of Logarithmic Functions

When working with logarithmic functions, determining the domain, or valid input values, is essential. For a logarithmic function like \( ln(x)\), the input, \(x\), must always be positive. This requirement stems from the mathematical rule that you cannot take the logarithm of a non-positive number.
Therefore, when solving logarithmic equations, any proposed solution (or value of \(x\)) must be checked to ensure it lies within the domain of the function. In our case, the domain is \(x > 0\).
Failure to ensure the potential solution is within this domain could lead to incorrect or undefined results, which is why this step is crucial in solving these types of problems.

Exponential Form Conversion

To solve equations involving logarithms, converting the logarithmic form to its exponential equivalent is often very useful. This process may simplify the equation greatly.
For example, the equation \( ln(x) = -\frac{1}{3}\) is in logarithmic form. To convert it, remember that if \( ln(b) = a\), it can be rewritten in exponential form as \(b = e^a\).
Applying this to our specific problem, we combine the base \(e\) and the logarithmic expression to find \(x\) as \(x = e^{-\frac{1}{3}}\).
The transformation from logarithmic to exponential form often makes the next steps in solving the equation more straightforward, particularly when calculating the numerical value of the variable.

Decimal Approximation

Once we have an expression in an exponential form, such as \(x = e^{-\frac{1}{3}}\), it may be necessary to find a decimal approximation for everyday use or to provide a final answer in a more accessible format.
Computing a decimal approximation involves using a calculator to ensure precision, especially since dealing with numbers like \(e\) (approximately 2.71828) can lead to lengthy decimals.
When using a calculator, it's important to consider the precision you need. In many cases, solutions are rounded to two decimal places for clarity and succinctness. Therefore, the calculated expression \(e^{-\frac{1}{3}}\) approximates to \(0.72\) when rounded to two decimal places.
These rounded results are often utilized in practical applications where exact numbers are not strictly necessary.