Problem 53 - Introductory and Intermediate Algebra for College Students 4th

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 value of \(x\) in the expression \(log _{4}(3 x+2)=3\) is \(x = \frac{62}{3}\) or approximately \(x=20.67\) to two decimal places.

Step 1: Write the Problem

First, write down the problem: \(log _{4}(3 x+2)=3\).

Step 2: Change to Exponential Form

Recall that logarithm and exponentiation are inverse operations. So the equation can be rewritten in exponential form: \(4^{3}=3 x+2\). Now, the log is no longer present.

Step 3: Solve for \(x\)

Perform algebraic operations to solve for \(x\). Thus, the equation \(4^{3}=3 x+2\) becomes \(64=3x+2\). Isolate \(x\) by subtracting 2 from both sides, giving \(62=3x\). Finally, divide by 3 to solve for \(x\), we get \(x = \frac{62}{3}\), which gives approximately \(x=20.67\) when rounded to two decimal places.