Solve the equation.
2log_2(x-5)+log_2(2)=5
To solve the equation 2log_2(x-5)+log_2(2)=5, we need to apply the properties of logarithms.
1. Start by combining the logarithms. Since log_2(2) = 1, we can rewrite the equation as:
2log_2(x-5) + 1 = 5
2. Now, we can simplify the equation further by multiplying 2 to log_2(x-5):
log_2((x-5)^2) + 1 = 5
3. Next, subtract 1 from both sides of the equation:
log_2((x-5)^2) = 4
4. Rewrite the equation in exponential form:
2^4 = (x-5)^2
5. Simplify the left side:
16 = (x-5)^2
6. Take the square root of both sides to eliminate the square:
±√16 = ±(x-5)
7. Solve for x by applying both the positive and negative square roots:
x-5 = ±4
8. Add 5 to both sides to isolate x:
For x-5 = 4:
x = 4 + 5 = 9
For x-5 = -4:
x = -4 + 5 = 1
Therefore, the solutions to the equation 2log_2(x-5)+log_2(2)=5 are x = 9 and x = 1.