Question

solve the logarithmic equation.\\(\log _{5}(2x - 9) = 2\\)\\(\\ \\)\\(\\ \\)select the correct choice below and, if necessary, fill in the answe\\(\bigcirc\\) a. \\(x = \square\\) (simplify your answer. use a comma to separate \\(\bigcirc\\) b. there is no solution.)

Explanation:

Step1: Convert log to exponential

Using the definition of logarithms: if $\log_b(a) = c$, then $b^c = a$. So for $\log_5(2x - 9) = 2$, we get $5^2 = 2x - 9$.

Step2: Simplify the exponential

Calculate $5^2 = 25$, so the equation becomes $25 = 2x - 9$.

Step3: Solve for x

Add 9 to both sides: $25 + 9 = 2x$, so $34 = 2x$. Then divide by 2: $x = \frac{34}{2} = 17$.

Step4: Check the domain

The argument of the logarithm $2x - 9$ must be positive. For $x = 17$, $2(17) - 9 = 34 - 9 = 25 > 0$, so it's valid.

Answer:

A. $x = 17$