Let's solve the equation step by step.
Given:
[tex]\[
\log_2(6x - 8) - \log_2 8 = 1
\][/tex]
1. Apply the property of logarithms [tex]\(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)\)[/tex] to combine the logarithms:
[tex]\[
\log_2\left(\frac{6x - 8}{8}\right) = 1
\][/tex]
2. Convert the logarithmic equation to its exponential form. Recall that if [tex]\(\log_b(A) = C\)[/tex], then [tex]\(b^C = A\)[/tex]. Here, [tex]\(b = 2\)[/tex], [tex]\(C = 1\)[/tex], and [tex]\(A = \frac{6x - 8}{8}\)[/tex]:
[tex]\[
2^1 = \frac{6x - 8}{8}
\][/tex]
3. Simplify the right-hand side of the equation:
[tex]\[
2 = \frac{6x - 8}{8}
\][/tex]
4. Clear the fraction by multiplying both sides of the equation by 8:
[tex]\[
2 \times 8 = 6x - 8
\][/tex]
[tex]\[
16 = 6x - 8
\][/tex]
5. Add 8 to both sides of the equation to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
16 + 8 = 6x
\][/tex]
[tex]\[
24 = 6x
\][/tex]
6. Divide both sides by 6 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{24}{6}
\][/tex]
[tex]\[
x = 4
\][/tex]
So, the value of [tex]\(x\)[/tex] that satisfies the equation is:
[tex]\[
\boxed{4}
\][/tex]
