y=logₐx is the inverse function of y=aˣ（symmetric about y=x）. Domain: x>0; asymptote: the y-axis（x=0）.
- a>1: monotonically increasing / 0<a<1: monotonically decreasing
- It always passes through the point (1,0)
Example ①
State the domain, asymptote, and a point on y=log₂(x−1)+3.
Answer: Domain x>1、asymptote x=1、when x=2, y=3 → point (2,3)
State the domain, asymptote, and a point on y=log₂(x−1)+3.
Answer: Domain x>1、asymptote x=1、when x=2, y=3 → point (2,3)
Key Point
- The argument must always be positive（note the domain） / increasing when a>1, decreasing when 0<a<1
Practice Problems
- Find the domain of y=log₃(2x+1).
- For y=log₂x, find the range of x for which y≥2.
- State the asymptote and the value at x=2 for y=log_(1/2)(x−1)+2.
Answers and Explanations
- Answer: x>−1/2
- Answer: x≥4
- Answer: Asymptote x=1、at x=2, y=2