Exponential Functions: Applications
𝑦 = 𝑎𝑏 𝑥
𝒂 is the initial amount
b is the growth or decay factor
x is the number of growth or decay periods
𝒚 is the future amount (amount available after x time periods.)
Words that may appear
in the problem
Growth Rate
Doubling
Tripling
Percentage Growth
Increasing by k%
Decay Rate
Half-Life
Percentage Decay
Decreasing by k%
Value of b
2
3
1+
𝑘
100
1
2
1−
𝑘
100
Note about a: If the problems is investigating the percent of a value that would be found after
x time periods, then 𝑎 = 1 (Meaning the starting value is 100%.)
Note about x: The number of growth or decay periods TIMES the length of each period gives
the total time for the problems.
1. A bacteria culture doubles every 15 minutes. How long will it take a culture of 20 bacteria to
grow to a population of 163 840?
2.
Deepa invests $1000 into an account that earns 6% interest per annum. How long will it take
her to double her money?
3. A balloon contains 3500mL of air. A small hole is causing a leak, and the balloon is losing 2% of
its air every 5 minutes. How long will it take for the balloon to reduce to 1000mL?
4. 230g of Londenium-13 has reduced to 10g. If the half-life of Lonendium-13 is 70 minutes,
determine the time that has passed since the 230g measurement was made.
5. 600g of a radioactive substance decays to 80g after 200 years. What is the half-life of the
substance?
Logarithmic Functions: Applications
Richter Scale (Intensity of Earthquakes)
The Richter Scale is used measure the intensity of earthquakes.
Richter Scale Earthquake
Measurement
5
7
9.5
True Intensity
105
107
109.5
FYI: There is not unit on the Richter Scale. The Intensity
is compared to a controlled explosion of TNT. So an
earthquake with a Richter Scale Intensity of 5 has a true
intensity of 105 , or 100000 times more intense than
that controlled explosion.
1. The city of Lowell experienced an earthquake that measured 6.2 on the Richter scale. The next
week, another earthquake was felt, which measured 7.1 on the Richter scale. How much more
intense was the second earthquake?
Decibels (Loudness of Sound)
Decibels are used measured using a logarithmic scale, but includes a factor of 10. So we need to take
the decibel measurement (dB) and divide by 10 to get the exponent for measuring true loudness.
Decibel Measurement
30
True Loudness
103
65
106.5
91
109.1
FYI: When created, the quietest sound that the human
ear can hear was given loudness of 0dB.
We now have mechanical equipment that can detect
quieter noises, so negative decibels do exist.
2. The sound of a leaf blower measures 105 on the decibel scale. A person shouting measures 80
dB. How many times more loud is the leaf blower?