To solve the equation [tex]\(\log_2 3 + \log_2 x = \log_2 5 + \log_2 (x-2)\)[/tex], follow these steps:
1. Combine the logarithms using the logarithm addition rule [tex]\(\log_b a + \log_b c = \log_b (a \cdot c)\)[/tex]:
[tex]\[
\log_2 (3x) = \log_2 (5(x-2))
\][/tex]
2. Since the logarithms on both sides of the equation have the same base and are equal, we can set their arguments equal to each other:
[tex]\[
3x = 5(x - 2)
\][/tex]
3. Now, solve the resulting linear equation:
[tex]\[
3x = 5x - 10
\][/tex]
4. Isolate [tex]\(x\)[/tex] on one side of the equation by first subtracting [tex]\(5x\)[/tex] from both sides:
[tex]\[
3x - 5x = -10
\][/tex]
5. This simplifies to:
[tex]\[
-2x = -10
\][/tex]
6. Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
x = \frac{-10}{-2}
\][/tex]
7. Simplify the division:
[tex]\[
x = 5
\][/tex]
Therefore, the solution to the equation [tex]\(\log_2 3 + \log_2 x = \log_2 5 + \log_2 (x-2)\)[/tex] is [tex]\(x = 5\)[/tex].
To ensure our solution is valid, we can verify that it satisfies the original equation:
- Substitute [tex]\(x = 5\)[/tex] back into the original equation:
[tex]\[
\log_2 3 + \log_2 5 = \log_2 5 + \log_2 (5-2)
\][/tex]
- Simplify both sides:
[tex]\[
\log_2 3 + \log_2 5 = \log_2 5 + \log_2 3
\][/tex]
- Recognize that both sides are indeed equal, confirming that [tex]\(x = 5\)[/tex] is correct.
So, the correct solution is [tex]\(x = 5\)[/tex].
