To solve the equation [tex]\(\log_2 11 - \log_2 \frac{x}{y} = \log_2 \frac{11}{9}\)[/tex], we can use properties of logarithms and basic algebraic manipulation. Here is the step-by-step solution:
1. Apply the properties of logarithms:
The logarithmic difference [tex]\(\log_b(a) - \log_b(c)\)[/tex] can be simplified to a single logarithm: [tex]\(\log_b(a/c)\)[/tex].
Therefore, the given expression:
[tex]\[
\log_2 11 - \log_2 \frac{x}{y}
\][/tex]
can be written as:
[tex]\[
\log_2 \left( \frac{11}{x/y} \right)
\][/tex]
Simplifying the argument of the logarithm:
[tex]\[
\log_2 \left( \frac{11 \cdot y}{x} \right)
\][/tex]
2. Set the logarithmic expressions equal to each other:
According to the given equation:
[tex]\[
\log_2 \left( \frac{11 \cdot y}{x} \right) = \log_2 \left( \frac{11}{9} \right)
\][/tex]
3. Equate the arguments:
Since the logarithms are equal, their arguments must also be equal:
[tex]\[
\frac{11 \cdot y}{x} = \frac{11}{9}
\][/tex]
4. Simplify the equation:
Divide both sides by 11 to isolate the fraction on the left side:
[tex]\[
\frac{y}{x} = \frac{1}{9}
\][/tex]
5. Solve for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
[tex]\[
y = \frac{x}{9}
\][/tex]
6. Determine specific values:
We can choose specific values for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy this equation. A straightforward solution is to set [tex]\(y = 1\)[/tex]. Thus:
[tex]\[
\frac{1}{x} = \frac{1}{9}
\][/tex]
7. Find the value of [tex]\(x\)[/tex]:
[tex]\[
x = 9
\][/tex]
So the specific values that satisfy the equation are:
[tex]\[
(x, y) = (9, 1)
\][/tex]
Hence, the solution to the given logarithmic equation is:
[tex]\[
(x, y) = (9, 1)
\][/tex]
