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Using Logarithms to solve B = A
1) Given the exponential equation Bx = A, use the change of base formula to solve for x
2) Given the logarithmic equation; log B A  x , use the change of base formula to solve for x
Solve:
3) 25x = 125
4) x4 = 12.8
5) 36x = 216
6) 4x = 100
7) 5x = .04
8) 10x = 125
9) 2x-2 = 128
10) 8x+3 = 500
11) 252x = 125
12) 34x+1 = 85
13) 3(2)x = 120
14) 4(6)x-1 = 864
15) 100(1.08)x = 1250
16) 650(1.025)x = 400
 .04 
17) 3001 

2 

 .18 
18) 1501 

12 

12 x
 1850
 .25 
19) 90001 

4 

4x
2x
 1.08 
 7500 20) 35001 

52 

 750
52 x
 20000
Properties of Logarithms
A. Rewrite as a single logarithm
B. Solve for the variable indicated
21) log x  4 log 2
1
22) log 5 u  log 5 8
3
23) log m  log 3  log 7
24) log y  log 35  log 7
25) log p  log 4  3 log 5
26) 4 log x  log 32  log 2
27) log 2 x  2 log 6  log 3
28) log 2
t
 2 log 2 5  3 log 2
3
Write each number as a decimal (do as many as possible without a calculator)
29) ln e14 
31) 5 ln e 
32) 5 ln e 2 
33) log 18 18 20  y
34) 7 log 3 3  8 log 3 3  y
34) log 8 100
34) ln 2.4  0.875
35) log e 3  y
37) e x  5.755
38) 2e x  12.8
Write in exponential form
33) ln 42  3.738
Write in logarithmic form
36) e 7  1097
39) Danny’s college savings are invested in a bond that pays an annual interest rate of 6.2%
compounded continuously. How long will it take the money to triple?
40) At what rate would you have to invest your money for it to double if it were
compounded continuously for 8 years?
41) At what rate would you have to invest your money for it to double if it were compounded
monthly for 2 years?
42) How long will it take $200 to grow to $1000 if it was invested @ an annual rate of 8%
compounded quarterly?
43) When a certain medical drug is administered to a patient, the number of milligrams
remaining in the patient’s bloodstream after t hours is modeled by D(t) = 50e-0.2t
A) Create a word model to describe the function D(t)
B) How many milligrams would remain in the blood after 3 hours?
C) Due to the toxicity of the drug and the possibility negative side effects, patients should not
take another dose until only 10% of the drug remains in the system. Approximately how long
after the first dose should you wait until taking a second dose?
44) Nancy wants to invest $4000 in savings certificates that bear an interest rate of 5.75%
compounded semi-annually. How long should she invest for if she wants to make $1200 in
interest?
45) Radioactive iodine decays exponentially in mass by 8.7% every day. If a sample began
with 20 grams, how long would it take until there were 10 grams remaining?
46) On January 1st 2002 the population of a country had a population of 4,100,000 people.
Since that time it has been growing at an annual rate of 3.5% (this is not
continuous/exponential). Assume that this has been and always will remain the same.
A) What was the population of the country 6 months later?
B) What was the population of the country 9 years later?
C) What will be the population of the country two decades later?
D) According to this model what would the population have been in 1970?
47) Find the missing variable and create a word problem to match each problem.
A) 4200 = 9000e15r
B) 4200 = 9000(r)15
r 

C) 42001  
 12 
180