Question

solve the logarithmic equation.\\(\log _{3}(5x - 3)=4\\)\\(\\ \\)\\(\\ \\)select the correct choice below and, if necessary, fill in the answer box to complete your choice\\(\\ \\)\\(\boldsymbol{\text{o a. }}\\ x = \square\\ \text{(simplify your answer. use a comma to separate answers as needed.)}\\)\\(\boldsymbol{\text{o b. }}\\ \text{there is no solution.}

Explanation:

Step1: Convert log to exponential form

Recall that if $\log_b(a) = c$, then $b^c = a$. For $\log_3(5x - 3) = 4$, we have $3^4 = 5x - 3$.

Step2: Calculate $3^4$ and solve for x

Calculate $3^4 = 81$. So the equation becomes $81 = 5x - 3$. Add 3 to both sides: $81 + 3 = 5x$, which is $84 = 5x$. Then divide both sides by 5: $x = \frac{84}{5} = 16.8$.

Step3: Check the domain

The argument of a logarithm must be positive. For $\log_3(5x - 3)$, we need $5x - 3 > 0$. Substitute $x = 16.8$: $5(16.8) - 3 = 84 - 3 = 81 > 0$, so it is valid.

Answer:

A. $x = \frac{84}{5}$ (or 16.8)