To solve the equation [tex]\(\log_6(13 - x) = 1\)[/tex], follow these steps:
1. Understand the given logarithmic equation: [tex]\(\log_6(13 - x) = 1\)[/tex]. This equation states that the logarithm of [tex]\(13 - x\)[/tex] with base 6 is equal to 1.
2. Convert the logarithmic equation to its exponential form. Recall that if [tex]\(\log_b(A) = C\)[/tex], then [tex]\(b^C = A\)[/tex]. Applying this to our equation:
[tex]\[
6^1 = 13 - x
\][/tex]
3. Simplify the exponential expression on the left-hand side:
[tex]\[
6 = 13 - x
\][/tex]
4. Solve for [tex]\(x\)[/tex] by isolating [tex]\(x\)[/tex] on one side of the equation. To do this, subtract 6 from both sides of the equation:
[tex]\[
6 - 6 = 13 - x - 6
\][/tex]
[tex]\[
0 = 13 - x - 6
\][/tex]
Simplify the right-hand side:
[tex]\[
13 - 6 = x
\][/tex]
5. Final simplified expression gives:
[tex]\[
x = 7
\][/tex]
Therefore, the solution to the logarithmic equation [tex]\(\log_6(13 - x) = 1\)[/tex] is:
[tex]\[
x = 7
\][/tex]
