Mathematical Investigations III
Name:
Sample Test - Logs
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1. Evaluate:
log5 (125)
æ 1ö
log 2 ç ÷
è 64 ø
2. Combine into a single logarithm then simplify. (Assume that all variables represent positive
numbers.)
a.
1
ln( x)  4 ln( xy )  2 ln( y ).
2
b. 3log5 ( 2 x  4)  4log5 ( x  1)  2log5 (25 x )
3. Solve for x: log 4 ( x)  log 4 ( x  6)  log 4 (16) .
4. Solve for x: 3·4 x+2 = 18.
5. Given the function f (x) = log3 (x + 2) - 5, find the inverse function f 1 ( x).
6. The half-life of a radioactive substance is 325 years. After how many years will the
substance decay to 60% of its original amount? Leave your answer in terms of the natural
log, ln( ).
7. Prove the following. Clearly show your steps to get full credit.
log a (b 4 )
log a4 (b) 
.
16
8.
Given p  log3 (2) and q  log3 (5) , express the following in terms of p, q if possible.
 16 
b. log 3 

 45 
a. log3  log81  log 2 8 
9. Solve for n in terms of a,b, and c: a 
b  cn
ac
Log Test p.1
SP12
Mathematical Investigations III
Name:
10. Graph each of the following functions. In each case label the scales on the x- and y- axes
clearly and label at least three points.
f ( x)  log 2 (2 x)
Domain =
Range =
f ( x)  log2 x 1
Domain =
Range =
f ( x)  log3 ( x  1)  2
Domain =
Range =
f ( x)  log 3 ( x 2 )
Domain =
Range =
Log Test p.2
SP12