The difficulty is with these log graphs that students can usually only
do it by rote. Very few have the slightest clue of what things like the
slope mean.
A question on my diagnostic test asks the student to make a reasonable
estimate of log (base 10) of 500. Any answer between 2 and 3 is
accepted.
OVERWHELMING majorities of students at nearly all institutions have no
idea of how to do this, including many students with 3 semesters of
calculus (including logarithmic functions). The most common wrong answer
is "50". Actually the most common is no answer.
I think a part of the blame is on calculus courses and texts that define
the logarithm by an integral. This leaves the students with no intuition
or understanding of what they are as exponents, and why they have the
properties they do. Defining the log this way is only appropriate AFTER
the students have a real understanding of and facility with logs as
exponents. From my data I would say that maybe 1 or 2 students in 100
have this knowledge on leaving high school.
J. Epstein