Logarithms
1. Find (if possible): A. log 1.035
D. ln  e 
 1 
B. log 4  
 64 
C. log 2  4 
E. log9  3
2. Graph y  ln x . State the domain, range, and include features such as asymptotes and intercepts.
3. Prove that y  log a  x  is proportional to y  log  x  . What is the proportionality constant?
4. Derive the formula for the tripling time for P  P0 a nt . What does this tripling time NOT
depend on? Do we need to make any assumptions about a and n ?
5. Consider S  D   0.159  0.118log  D  . S is the slope of a beach and D is the average diameter
(in mm) of the sand particles on the beach. Suppose a particular beach rises 9 meters for every 100
meters inland. What type of sand (what size) would you expect to find on that beach? If you
actually found sand with an average diameter of 0.8 mm instead, what do you think happened?
Note: D  0.125 mm indicates fine sand, D  0.5 mm indicates course sand, and D  4 mm
Indicates pebbles.
6. You and a friend have just purchased cars. The value of your car is $34,000 and will depreciate
17% each year. The value of your friend’s car is $16,500 and will depreciate 12% each year. You
agree to exchange cars when their values are equal. Determine how long you will wait (to the
nearest day) and what the value of the cars will be at that time. Would your answer change if the
depreciation rates were continuous instead of annual rates? What would your depreciation rate
have to be in order for the values of the cars to match at the end of 7 years?
7. Graph y  7
log7 x
. Include the domain.
8. Graph y  ln e x . Include the domain.