To find the value of [tex]\( x \)[/tex] that satisfies the equation
[tex]\[
\log_2(6x - 8) - \log_2(8) = 1
\][/tex]
we can follow these steps:
1. Combine the logarithmic terms using the properties of logarithms. Specifically, the difference of logarithms can be expressed as the logarithm of a quotient:
[tex]\[
\log_2(6x - 8) - \log_2(8) = \log_2 \left( \frac{6x - 8}{8} \right)
\][/tex]
2. Simplify the equation by setting the argument of the logarithm equal to the number on the right-hand side (because if [tex]\(\log_2(A) = B\)[/tex], then [tex]\(A = 2^B\)[/tex]):
[tex]\[
\log_2 \left( \frac{6x - 8}{8} \right) = 1
\][/tex]
3. Recall that [tex]\( \log_2(y) = 1 \)[/tex] implies [tex]\( y = 2 \)[/tex]. Therefore, we set the argument equal to 2:
[tex]\[
\frac{6x - 8}{8} = 2
\][/tex]
4. Solve the equation for [tex]\( x \)[/tex]. Start by clearing the fraction by multiplying both sides by 8:
[tex]\[
6x - 8 = 16
\][/tex]
5. Next, add 8 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
6x = 24
\][/tex]
6. Finally, divide both sides by 6 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 4
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] that satisfies the given logarithmic equation is
[tex]\[
\boxed{4}
\][/tex]
