MAC 1140 Section 4.3 - Properties of Logarithms
Multiplication property:
Division property:
Exponent property:
log MN = log M + log N
M
log
= log M - log N
N
log MN = N log M
A. Rewrite as a single logarithm.
1. log 2 + log y
2. log 12 – log 4
3. ln x9 - ln x3
4. log3x + log3(x-4)
5. log2(18) – log2(6)
6. ln(x2 – 49) – ln(x + 7)
B. Rewrite each expression as a sum or difference of logarithms.
5
7. log  
 y
8. log5(4x)
 x 3
9. log 

 x 
10. log3(8m)
k
11. ln  
4
12. log x(x-3)
C. Use the properties of logarithms to write the following as sums or differences of logarithms of
single variables.
13. log (x+2)5
14. log t3
15. log
17. log 4x5
18. ln x3y2
19. log (wm)2
20. ln
23. log2(5xy)
 5m 
24. ln 

 7b 
21. log
ab
xy
22. log
5
3m
x
16. ln
3
y
x
y
D. Rewrite each expression as a single logarithm. Take care of any exponents first.
25. log2(7) + 4log2(y)
26. log(y) + 3log(y)
27. 2 log x - 3 log (x +1)
28. ln (2) + ln (5) – ln(w)
29. 2logw + log5 – 3logw
30. lnx + 2ln y – ln4 – ln 2
31. 3ln x –
33.
1
ln (x+1) - 3ln 5
2
32. 2log3(4) - log3(X) +
1
1
ln(x) – ln(y) - ln(k) + ln(3)
3
2
34. 2ln(m) – ln(r) + 3ln(m) -
E. SIMPLIFY—Do not use a calculator. Use
b logb x  x
2. ln ew
3. log 5 5 x  2
4. log 6 6 x
6. 10logx
7. eln t
8. 7log 7 x
9. 5log 5 ( x  1)
1. log3 (4.78)
1
ln(y)
2
logb b x  x
1. log 3 3x
F. Change of base formula: logba =
1
log3y
2
2
5.
log b b x
10. blog b x
log a
log(50)
. Example: log350 =
= 3.5609
log b
log(3)
2. log1.2 (13.7)
3. log1.05 (3.66)
G. Find an approximate rational solution to each equation. Round answer to four decimal places.
(Use your calculator; Let Y1= left side of equation; Y2 = right side of equation. Graph using
Zoom6. If point of intersection is not in window, adjust window until it is in window. Use 2nd,
Trace(CALC), 5 for intersect. The x-coordinate of point of intersection is solution.)
1. 3x = 8
2. 5x = 15
3. (1.09)x = 8
HW: Page 296 Problems: 1-11 odds, 15, 23-27 all, 37-43 all, 49- 56 all;
Use Calculator on 59, 61, 62, 67, 69,71
4. 3.1x = 13
(074)