Final answer:
To solve the equation log₂(x-1) = -1 algebraically, we can rewrite the equation in exponential form to get 2^{-1} = x - 1 and solve for x to get x = 1.5.
Step-by-step explanation:
The student has presented an equation that involves a logarithm: log₂(x-1) = -1. To solve for x, we must first understand that the log base 2 of a number equals -1 when that number is 1/2.
We can rewrite the equation in exponential form, transforming the logarithmic equation into a basic algebraic one. The base of the log (which is 2) becomes the base of the exponent, and the -1 becomes the exponent on the right side of the equation, like so:
To solve for x now that we have an exponential equation, we calculate 2-1 which is 1/2 or 0.5, and then add 1 to both sides to isolate x:
- 0.5 = x - 1
- 0.5 + 1 = x
- x = 1.5
Therefore, the exact solution to the equation log₂(x-1) = -1 is x = 1.5.