Problem 57 - 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. $$\log _{3}(x+4)=-3$$

Step-by-Step Solution

Answer

The solution to the logarithmic equation \(\log _{3}(x+4)=-3\) is \(x=-3.96\), correct to two decimal places.

Step 1: Transform from Logarithmic to Exponential Form

Since \(\log _{3}(x+4)=-3\) is in logarithm form, it has to be transitioned into exponential form. According to log properties, it can be rewritten as \(3^{-3}=x+4\).

Step 2: Solve for \(x\)

Resolve \(3^{-3}=x+4\). Evaluating \(3^{-3}\) is 1/27, so we get \(\frac{1}{27}=x+4\). Thereafter, we will isolate \(x\) by subtracting 4 from each side, giving \(x=\frac{1}{27}-4\). Then the fraction is simplified to give the exact answer.

Step 3: Check The Domain

We have to ensure the result is in the domain of the original logarithmic function, that is \(x+4\) must be greater than 0. After substituting \(x=\frac{1}{27}-4\) into \(x+4\), it is found to be a positive number, hence, the result is valid.

Step 4: Decimal Approximation

Lastly, the exact answer is approximated to two decimal places using a calculator, resulting in \(x=-3.96\)