To solve the equation [tex]\(\log_7(5 + 11x) = 2\)[/tex], follow these steps:
1. Rewrite the Logarithmic Equation in Exponential Form:
The equation [tex]\(\log_7(5 + 11x) = 2\)[/tex] implies that [tex]\(7^2 = 5 + 11x\)[/tex].
2. Compute the Exponent:
Calculate [tex]\(7^2\)[/tex]:
[tex]\[
7^2 = 49
\][/tex]
So, the equation becomes:
[tex]\[
49 = 5 + 11x
\][/tex]
3. Isolate the term containing [tex]\(x\)[/tex]:
Subtract 5 from both sides of the equation to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
49 - 5 = 11x
\][/tex]
Simplify the left-hand side:
[tex]\[
44 = 11x
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides of the equation by 11 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{44}{11}
\][/tex]
Simplify the fraction:
[tex]\[
x = 4
\][/tex]
Therefore, the solution to the equation [tex]\(\log_7(5 + 11x) = 2\)[/tex] is:
[tex]\[
x = 4
\][/tex]
So, the correct answer to the question is:
[tex]\[
\boxed{4}
\][/tex]
