Problem 73 - 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. $$2 \log _{3}(x+4)=\log _{3} 9+2$$

Step-by-Step Solution

Answer

The solution for the given logarithmic equation is \(x = -1\). The value \(x = -7\) is rejected as it's not in the domain of original logarithmic expressions.

Step 1: Simplification

Simplify the right side of the equation where possible. In our case, \( \log_3 9 = 2\), hence the equation becomes: \(2 \log_3 (x+4) = 2 + 2 \). Subtracting 2 from both sides, we get \(2 \log_3 (x+4) = 2 \)

Step 2: Apply logarithmic rules

Next, we apply the logarithmic rule and simplify: \(a \log_b c = \log_b (c^a)\). So, our equation becomes: \( \log_3 (x+4)^2 = 2 \)

Step 3: Change to exponential form

We can change the logarithm equation into an exponential one to solve for x: \(3^2 = (x+4)^2\). Calculating, we get \(9 = (x+4)^2\).

Step 4: Solve for x

Solve this equation by taking the square root on both sides: \(x + 4 = \pm \sqrt{9}\) yields \(x = \pm 3 - 4\).

Step 5: Check for domain validity

The \(x\) obtained from above needs to be cross-checked whether it fall under the domain of original logarithmic expression (\(x+4 > 0\)). For \(x = -1\), the domain is satisfied (-1+4 > 0). However for \(x = -7\), it's not (-7+4 < 0). Hence, x = -7 is rejected.

Step 6: Decimal approximation

The exact solution is \(x = -1\). No need for decimal approximation as it already is a decimal number.