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Applications of Exponents and Logs Solutions

Note

It is not possible to cover every type of exponential and logarithmic application because of the number of ways you can use them. Therefore, you need to use your problem-solving skills and apply your knowledge.

Each application of exponents and logarithms has its own formula and variables . 
To start, remember to identify the known and unknown values based on the formula provided. 
After solving, be sure to label your answer. 
Scientific notation is often used to write the final answer because exponents and logarithms increase or decrease rapidly.

Half-Life Application

Half-life is the time it takes for a substance to decrease to half of its original value, which can be true of radioactive materials, bacteria, medicine, etc.  
The general formula for a half-life is: At=A012th 
- $A ₀ :$ initial quantity
-  t: time , in years 
- h: half-life
- A(t): quantity remaining at time t

Continuous Compounding Application

The formula for continuous compounding events is: $y = P e^{r t}$

It can be used for growth or decay when the rate is continuously compounding.

Example 1

A new type of battery has a half-life of 5 years. To determine total life of the battery, lab technicians set up an experiment that starts with a fully charged battery (100%) and runs until the power remaining is 10%. How many years is the battery expected to last?

$P (t) = P_{0} {(\frac{1}{2})}^{\frac{t}{h}}$

Plan

Define known variables

Substitute values into the formula

Take the log of both sides

Simplify logs

Solve

$\begin{array}{rcl} 0.1 & = & 1 {(\frac{1}{2})}^{\frac{t}{5}} \\ \log 0.1 & = & \frac{t}{5} \log (\frac{1}{2}) \\ \log 0.1 & = & \frac{t}{5} (\log 1 - \log 2) \\ - 1 & = & \frac{t}{5} (0 - \log 2) \\ - 1 & = & \frac{t}{5} (- \log 2) \\ (\frac{5}{\log 2}) (1) & = & \frac{t \log 2}{5} (\frac{5}{\log 2}) \\ t & = & \frac{5}{\log 2} \\ t & = & 16.61 years \end{array}$

Implement

P(t): power remaining at time t $10 % = 0.1$

$P ₀ :$ initial power $100 % = 1$

t: time (in years) ?

h: half-life 5 yrs

$\log 1 = 0 \log 0.1 = - 1$

The life of the battery will be approximately 16.61 years.

Example 2

A population ecologist is studying the introduction of an invasive species of bugs into an environment with no predators. The initial number of bugs was estimated to be 400. If the ecologist estimates a population of 50,000 after 46 weeks, what is the continuous growth rate? Round to the nearest tenth of a percent.

$y = P e^{r t}$

$P = 400 t = 46 y = 50 000$

$\begin{array}{rcl} 50 000 & = & 400 e^{r \cdot 46} \\ 125 & = & e^{46 r} \\ \ln 125 & = & \ln e^{46 r} \\ \frac{\ln 125}{46} & = & \frac{46 r}{46} \\ r & = & 0.1049 \end{array}$

The bug population has a growth rate of approximately 10.5%.

Example 3

The pH of a solution measures its acidity or alkalinity. The pH is defined using the equation $pH = - \log [H +]$ and is measured in moles per liter.

The pH level for a section of the Schuylkill River near the city of Philadelphia, PA is 6.5. Determine the hydrogen ion concentration of the river water in moles per liter.

$\begin{array}{rcl} pH & = & - \log [H +] \\ 6.5 & = & - \log [H +] \\ - 6.5 & = & \log [H +] \\ H + & = & 10^{- 6.5} \\ = & 0.0000003162 \\ H + & = & 3.16 \times 10^{- 7} \end{array}$

The pH of the Schuylkill River is $3.16 \times 10^{- 7}$ moles/liter.