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1. Write as a single logarithm and evaluate if possible.
1
log 6 9  log 6 5
2
d) 2 log 5  log 4
a) log 5 2  log 5 3  log 5 4
b)
c) 2 log 3 6  2 log 3 2
1
e) log 2 48  log 2 27
3
g) log 8 80  log 8 5
1
i) (2 log M  log N  log P)
3
f) log M  3 log N
h) log A  2 log B  3 log C
2. Solve for the unknown
b) log 9 (2 y  1)  log 9 y 
a) log( x  3)  log( x  2)  1
1

b) log 2 ( x 2  2)  log 2  x  5   1
2

e) log 4 (2x  1)  log 4 ( x  2)  1
1
2
d) log 6 ( x  1)  log 6 x  1
f) log 2 ( x 2  8)  log 2 x  log 2 6
3. Change each logarithm to base 10 and evaluate to FOUR decimal places, then check your answers.
a) log 3 10
b) log 5 19
c) log 7 22
d) log 3.5 42
e) log 9 5
f)
log 4 0.03
4. Solve each exponential equation.
5.
a) 2 t  7
b) 0.4   5
c) 3 x  52
d) 8 x  3 5
g) 1.6 2 x 7  2.6
e) 5 2 x 1  17
h) 5000  3000(1.09) n
f) 10 3 x 5  20
i) 223  130(1.07) t
t
a) The population of a certain colony of bacteria quadruples every hour. If there are 2000 present
initially, how long will it take to grow to 14000?
b) The population, P, of Alberta has grown an average of 1.4% since 1981 when there were 2.28
million people. If n represents the number of years since 1981, and assuming this rate of
growth continues, in what year will the population be 4 million?
c) Repeat a) for a population that quadruples every 3 hours.
d) You invested $1000 at a fixed annual interest rate at 8%. Your friend Natalie invested $1000
at 16% every 2 years. Determine which investment will take longer to grow to $5000 and by
how long.
e) Carbon-14 (C14) found in the skeletal remains of animals, decays by 50% in approximately
5700 years. In one particular specimen, archaeologists estimated that the animal originally
contained 2.7 mg of C14, but now contains 0.0027 mg. How old are the remains to the nearest
thousand years?
f) For every 5 metres, a diver descends underwater, the intensity of blue light is reduced by
13.1%. At what depth is the blue light reduced to half?
g) A colony of bees increases by 25% every three months. If there are 2620 bees initially, how
long will it take to the nearest month for the colony to grow to 10 000 bees?
h) The half-life of Iodine-131 is 8.1 days. If you have 125 mg of Iodine-131 initially, how long
will it take to decay to 20% of the amount?
i) A cup of coffee is cooling from an initial temperature of 95 C and loses 7% of its heat every
2 minutes. How long will it take the cup of coffee to cool to the room temperature of 25 C
1. a) Scientists have discovered that people perceive a 40 dB to be twice as loud as a 20 dB sound.
How many more times intense is this?
b) How many times more intense is an earthquake measuring 7.1 on the Richter scale than one
measuring 6.9?
c) Another formula for the Richter scale is based upon the energy, E, in kilowatt hours, released by
the earthquake, R  0.67 log( 0.37 E )  1.46 where R is the Richter magnitude. A 1906
earthquake in San Francisco measured 7.90 on the Richter scale. How much energy was released
by this earthquake?
d) What is the total combined decibel level of two stereos, each playing the same music
simultaneously at 62 dB. What would it be for 3 stereos?
e) The decibel level of a new sports car is 80 dB as it accelerates. What would be the decibel level
for four identical cars? How many more times intense is this than the single car’s sound?
f) The observed brightness of stars is classified by magnitude. Two stars can be compared by giving
their magnitude difference d, or their brightness ratio, r. The numbers d and r are related by the
equation, d  2.5 log r . Comparing a first magnitude star with a sixth magnitude star gives d = 61=5. How many times brighter is the sixth magnitude star?