To solve the equation [tex]\(\log_6 22 = \log_6 (5x - 8)\)[/tex] for [tex]\(x\)[/tex], follow these steps:
1. Understand the Logarithmic Equations:
Since the logarithms on both sides of the equation have the same base (base 6), we can equate the arguments of the logarithms directly. This means:
[tex]\[
22 = 5x - 8
\][/tex]
2. Isolate [tex]\(5x\)[/tex]:
To solve for [tex]\(x\)[/tex], first isolate the term containing [tex]\(x\)[/tex]. Add 8 to both sides of the equation:
[tex]\[
22 + 8 = 5x
\][/tex]
Simplifying the left-hand side gives:
[tex]\[
30 = 5x
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Divide both sides of the equation by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{30}{5}
\][/tex]
Simplifying this gives:
[tex]\[
x = 6
\][/tex]
Thus, the solution to the equation [tex]\(\log_6 22 = \log_6 (5x - 8)\)[/tex] is:
[tex]\[
x = 6
\][/tex]
Hence, the correct answer is:
A. [tex]\(x = 6\)[/tex]
