Question

solve the following logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expression. give the exact answer. \\(\log_{2}(3x + 4) = 6\\) \\(\vdots\\) rewrite the given equation without logarithms. do not solve for x. solve the equation. select the correct choice below and, if necessary, fill in the answer box to complete your choice. \\(\bigcirc\\) a. the solution set is \\(\boxed{}\\). (type an integer or a simplified fraction.) \\(\bigcirc\\) b. there are infinitely many solutions. \\(\bigcirc\\) c. there is no solution.

Explanation:

Step1: Recall logarithm definition

The logarithmic equation $\log_{b}a = c$ can be rewritten in exponential form as $b^{c}=a$. Here, $b = 2$, $a=3x + 4$, and $c = 6$.

Step2: Apply the definition

Using the definition, we rewrite $\log_{2}(3x + 4)=6$ as $2^{6}=3x + 4$.

Step3: Solve for x

First, calculate $2^{6}=64$. So the equation becomes $64=3x + 4$. Subtract 4 from both sides: $64 - 4=3x$, which simplifies to $60 = 3x$. Then divide both sides by 3: $x=\frac{60}{3}=20$.

Step4: Check domain

The domain of $\log_{2}(3x + 4)$ requires $3x+4>0$. For $x = 20$, $3(20)+4=64>0$, so $x = 20$ is in the domain.

Answer:

The solution set is $\{20\}$ (corresponding to option A with the value 20).