Four things that trip people up
1. The asymptote is a fence, not part of the curve. The graph of y = logb(x − h) + k gets as close to the line x = h as you like and never touches it. Do not draw the curve crossing it, and do not draw a dot where they seem to meet.
2. There is no log of 0 and no log of a negative. The domain is x − h > 0, that is x > h. Ask why: log2(−8) would mean 2? = −8, and a positive base raised to any real power stays positive. Whenever you solve a log equation, check every answer against x > h and throw out the ones that fail — those are extraneous solutions.
3. The curve never levels off. It looks flat on the right, but it keeps climbing forever — there is no horizontal asymptote and no maximum. log2 1000 ≈ 9.97, and log2 1000000 ≈ 19.93. A thousand times bigger buys you only ten more units of height.
4. It is not a square-root curve. √ x starts at the point (0, 0) and stops. logb x has no starting point — it plunges toward −∞ as x approaches 0 from the right. Same lazy rise, completely different left end.