Problem 77 - 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. $$\log (x+4)=\log x+\log 4$$

Step-by-Step Solution

Answer

The exact solution is \(x = \frac{4}{3}\) and the decimal approximation for the solution is \(x \approx 1.33\).

Step 1: Combine the Logs on the Right

To start, apply the product rule of logarithms to \(\log x + \log 4\) on the right side of the equation. This simplifies the equation to \(\log (x+4) = \log (4x)\). This rule states that the log of a product is the sum of the logs of its factors.

Step 2: Use the Rule of Logarithms

Given that \(\log a = \log b\) implies \(a = b\), our equation \(\log (x+4) = \log (4x)\) can be simplified to \(x+4 = 4x\).

Step 3: Solve for x

Next, solve the equation \(x+4 = 4x\) for \(x\). To do this, subtract \(x\) from both sides to get \(4 = 3x\). Then, divide both sides by 3 to solve for \(x\), resulting in \(x = \frac{4}{3}\).

Step 4: Verify the Domain of the Solution

In the original logarithmic expressions \(\log (x+4)\) and \(\log x\), \(x\) must be greater than 0 because logarithm of non-positive numbers isn’t defined. Thus substitute the found \(x\) into the expressions and ensure they are defined. We find that the expressions \(\log\left(\frac{4}{3} +4 \right)\) and \(\log\left(\frac{4}{3}\right)\) are defined. Hence \(x = \frac{4}{3}\) is within the permissible domain.

Step 5: Decimal Approximation

By approximating \(x\) to two decimal places using calculator, we get \(x \approx 1.33\).