Problem 59 - 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. $$\log _{4}(3 x+2)=3$$

Step-by-Step Solution

Answer

The exact solution to the equation is \(x = \frac{62}{3}\), and the decimal approximation is \(x \approx 20.67\).

Step 1: Convert the logarithmic equation to exponential form

Using the property of logarithms that says \(\log_b a = c\) is the same as \(b^c = a\), the equation \(\log_{4}(3x+2) = 3\) can be rewritten in exponential form as \(4^3 = 3x + 2\).

Step 2: Solve for \(x\)

Simplifying the equation from step 1 yields \(64 = 3x + 2\). You can subtract 2 from both sides to get \(62 = 3x\), and then divide by 3 to isolate \(x\), giving you \(x = \frac{62}{3}\).

Step 3: Check if \(x\) is in the domain of the original logarithmic expression

The original logarithmic expression is \(\log_{4}(3x+2)\), which is undefined for values of \(x\) that make \(3x + 2 \leq 0\). Substituting \(x = \frac{62}{3}\) into \(3x + 2\) gives a positive result therefore \(x = \frac{62}{3}\) is in the domain of the original logarithmic expression.

Step 4: Obtain decimal approximation

Using a calculator, \(x = \frac{62}{3}\) approximates to \(x \approx 20.67\), correct to two decimal places.