Wize High School Algebra II Textbook (Common Core) > Logarithmic Functions

Applications of Logarithmic Functions

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Applications of Logarithmic Functions
Compound Interest
A=P(1+in)nt\boxed{A=P\Bigg(1+\dfrac{i}{n}\Bigg)^{nt}}A=P(1+ni​)nt​  A=Final Amount ($)P=Initial Amount ($)i=Interest raten=Compound periodst=Time elapsed\begin{array}{rcl}
A&=&\text{Final Amount (\$)}\\\\
P&=&\text{Initial Amount (\$)}\\\\
i&=&\text{Interest rate}\\\\
n&=&\text{Compound periods}\\\\
t&=&\text{Time elapsed}
\end{array}APint​=====​Final Amount ($)Initial Amount ($)Interest rateCompound periodsTime elapsed​

Richter Scale
R=log⁡(II0)\boxed{R=\log\Bigg(\dfrac{I}{I_0}\Bigg)}R=log(I0​I​)​     R=Richter scale magnitudeI=IntensityI0=Threshold Intensity\begin{array}{rcl}
R&=&\text{Richter scale magnitude}\\\\
I&=&\text{Intensity}\\\\
I_0&=&\text{Threshold Intensity}
\end{array}RII0​​===​Richter scale magnitudeIntensityThreshold Intensity​

ph Scale
pH=−log⁡H+\boxed{\text{pH}=-\log_{}\text{H}^+}pH=−log​H+​   H+=Hydrogen Ion Concentrationph=ph unit\begin{array}{rcl}
\text{H}^+&=&\text{Hydrogen Ion Concentration}\\\\
\text{ph}&=&\text{ph unit}
\end{array}H+ph​==​Hydrogen Ion Concentrationph unit​

Decibel Scale
 
dB=10log⁡(II0)\boxed{dB=10\log\Bigg(\dfrac{I}{I_0}\Bigg)}dB=10log(I0​I​)​     dB=Decibel unitI=Final SoundI0=Threshold Sound\begin{array}{rcl}
dB&=&\text{Decibel unit}\\\\
I&=&\text{Final Sound}\\\\
I_0&=&\text{Threshold Sound}
\end{array}dBII0​​===​Decibel unitFinal SoundThreshold Sound​

Example: Applications of Logarithmic Functions
Mateo has $4500 to invest and is deciding between two options:
Option A: invest in an account that gives 3.25% compounded bi-weekly (every 2 weeks)
Option B: invest in an account that gives 3.20% compounded semi-monthly (twice a month)
How long will it take to double his investment with each option? Which is the better investment?

 Option A:                                                                          Option B: 

P=4500i=0.0325n=26\begin{array}{rcl}
P&=&4500\\\\
i&=&0.0325\\\\
n&=&26
\end{array}Pin​===​45000.032526​                                                         P=4500i=0.032n=24\begin{array}{rcl}
P&=&4500\\\\
i&=&0.032\\\\
n&=&24
\end{array}Pin​===​45000.03224​ 

A=P(1+in)nt9000=4500(1+0.032526)26t2=(1.00125)26tlog⁡2=26tlog⁡1.00125t=log⁡226log⁡1.00125t≈21.34 years\begin{array}{rcl}
A&=&P\Bigg(1+\dfrac{i}{n}\Bigg)^{nt}\\\\
9000&=&4500\Bigg(1+\dfrac{0.0325}{26}\Bigg)^{26t}\\\\
2&=&(1.00125)^{26t}\\\\
\log2&=&26t\log1.00125\\\\
t&=&\dfrac{\log2}{26\log1.00125}\\\\
t&\approx&21.34~\text{years}
\end{array}A90002log2tt​=====≈​P(1+ni​)nt4500(1+260.0325​)26t(1.00125)26t26tlog1.0012526log1.00125log2​21.34 years​                      A=P(1+in)nt9000=4500(1+0.03224)24t2=(1.00133)24tlog⁡2=24tlog⁡1.00133t=log⁡224log⁡1.00133t≈21.73 years\begin{array}{rcl}
A&=&P\Bigg(1+\dfrac{i}{n}\Bigg)^{nt}\\\\
9000&=&4500\Bigg(1+\dfrac{0.032}{24}\Bigg)^{24t}\\\\
2&=&(1.00133)^{24t}\\\\
\log2&=&24t\log1.00133\\\\
t&=&\dfrac{\log2}{24\log1.00133}\\\\
t&\approx&21.73~\text{years}
\end{array}A90002log2tt​=====≈​P(1+ni​)nt4500(1+240.032​)24t(1.00133)24t24tlog1.0013324log1.00133log2​21.73 years​

Therefore, Option A is the better option. 

Practice: Applications of Logarithmic Functions
The hydrogen ion concentration of a certain liquid tests to be 0.0005. Is this liquid acidic or alkaline?

Answer

I don't know

Practice: Applications of Logarithmic Functions
George has $4000 to invest into an RRSP account that gives 12.5% annually. In 30 years, George plans on withdrawing his savings to use for his retirement. Unfortunately, when he withdraws his savings, he gets taxed at 
10% for withdrawals up to $5000
20% for withdrawals between $5000 and $15000
30% for withdrawals over $15000. 

a. How long will it take George to reach $5000? (Round to nearest hundredth)

b. If George withdraws all of his money in 30 years, how much will have to live off of for his retirement? (to the nearest dollar)

I don't know

Practice: Applications of Logarithmic Functions
Edmund and Gracie and tracking a substance and its half-life. They began with 300mg of the substance and after 15 days, they were left with 10mg. What is the half-life of this substance?

To the nearest hundredth (ignore units)

I don't know

Extra Practice

Applications of Logarithmic Functions

Practice: Applications of Logarithmic Functions
A car depreciates in value by 5% annually the moment it is driven off the lot. Geraldine buys a brand new Tesla for $50,000\$50,000$50,000 (there was a killer sale going on!) She wants to sell the car in 15 years. How much money will Geralinde lose on the sale of her car?

Applications of Logarithmic Functions

Practice: Applications of Logarithmic Functions
The half-life of a certain bacteria is 1.5 seconds. If the substance began at1000μg1000\mu{g}1000μg, how long until there 50μg50\mu{g}50μg left?

Applications of Logarithmic Functions

Practice: Applications of Logarithmic Functions
Atticus puts $4000 into his RDSP every year for 10 years. The government gives Atticus an extra $2500 at the end of each year since his income is below $30,000/year.
At the end of 10 years, Atticus will take half the money from his RDSP and invest it into a low-risk mutual fund that gives 1.5% back annually. If Atticus withdraws this money in another 20 years, how much money does Atticus have from his mutual fund?

Applications of Logarithmic Functions

Practice: Applications of Logarithmic Functions
Mya has $20,000 in money to invest into an RESP account that gives 3% annually. In 15 years, Mya plans on withdrawing her savings to use for her education. Her education will cost $35,000. 

Applications of Logarithmic Functions

Practice: Applications of Logarithmic Functions
The sugar concentration of a certain liquid tests to be 0.98. Is this liquid acidic or alkaline?

Applications of Logarithmic Functions

Practice: Applications of Logarithmic Functions
The nitrogen concentration of a certain liquid tests to be 0.008. Is this liquid acidic or alkaline?