Math-3 Lesson 4-8 Logarithmic and Exponential Modeling

What you’ll learn about Orders of Magnitude and Logarithmic Models Quantities and units of measure Newton’s Law of Cooling Measuring earthquakes Measuring pH … and why The Richter scale, pH, and Newton’s Law of Cooling, are examples of real world phenomena that are modeled by logarithmic and exponential functions.

The common logarithm of a positive quantity is its order of magnitude. The exponent of 10 that is equivalent to the number is the order of magnitude of the number.

What is the order of magnitude of the following number? The exponent of 10 that is equivalent to the number is the order of magnitude of the number. Usually, we are only interested in the difference of the order of magnitude between two numbers.

Finding the difference of the order of magnitude between two numbers. difference : The order of magnitude difference between the two numbers is: About 3 orders of Mag. Which means

Your turn: Find the order of magnitude difference between the distance Mercury is from the sun and the distance Pluto is from the sun. The common logarithm of a positive quantity is its order of magnitude. Distance: Mercury to the sun Distance: Pluto to the sun About 3 orders of Mag.

Your Turn: What is the order of magnitude difference between a gram and a kilogram? What is the order of magnitude difference between a person’s $100,000 salary and the $1 Trillion the government spent that it didn’t have last year (and borrowed from you)?

Vocabulary Quantity: something in the real world that can be measured. Height Weight Temperature Rate: (a ratio of quantities) becomes a new quantity.

Vocabulary Height is measured in units of inches (in.), feet (ft.), miles (mi.), etc. Unit of Measure: the “ruler” used to measure how “big” a quantity is. Weight is measured in units of pounds (lbf.), kilograms (kg.), etc.

Speed is measured in units of miles per hour, kilometers per hour, feet per second, etc. Pressure is measured in units of newtons per square meter, pounds-force per square inch, etc.

Graphs oftentimes compare quantities. Each axis is labeled with the quantity (with the units of measure for the quantity in parentheses).

Base 2 Exponential Function We say this function has a growth factor of 2. Base ‘e’ Exponential Function We say this function has a growth rate of 2.

Suppose boiling water (100 ⁰ C) is taken off the stove to cool. Your turn: draw a graph of what you think the temperature will look like as time passes by. Label the x-axis and y-axis of your graph with the quantity that the axis represents. Your turn: Label the x-axis and y-axis of your graph with the units of measure (in parentheses) for the quantity the axis represents. Does the temperature go down forever? At what temperature does it start and end up at? Will it take hours, or minutes, or seconds to cool down?

When we combine “math” with measurements in the real world, we are concerned about units! Suppose boiling water (100 ⁰ C) is taken off the stove to cool. Temperature ( ⁰ C) ( ⁰ C) Time (min.)

Temp ( ⁰ C) Time (min.) Initial temperature Final temperature Cool down rate Exponents cannot have any units. What are the units of cool down rate?

-0.25*t has no units (they cancel out) Cool down rate Exponents cannot have any units. “t” has what units? -0.25*t (min) -0.25*(?)*t (min) That’s why we call “k” (coefficient of the exponent variable) a rate.

Newton’s Law of Cooling A high temperature item will cool off in a lower temperature medium in which it is placed. This cooling off process can be modeled by the following equation. Temperature (as a function of time) Surrounding (final) Temperature Initial Temp of the object Cooling rate Time

Newton’s Law of Cooling is the temperature of the surrounding medium is the temperature of the surrounding medium is the original temperature of the object is the original temperature of the object Isaac Newton discovered that a heated object will cool at a rate Isaac Newton discovered that a heated object will cool at a rate that is dependent upon the tempurature of the surrounding that is dependent upon the tempurature of the surrounding medium. The object’s temperature can be modeled by: medium. The object’s temperature can be modeled by: is time is time is a “fudge factor” for cooling time is a “fudge factor” for cooling time

Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C? T(5) = 55 We must first calculate “k” the “fudge factor” for this specific situation. situation. Isolate the variable

A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C? T(5) = 55 Now we know the complete modeling equation we can find the time to reach 25 degrees  T(t) = 25 the time to reach 25 degrees  T(t) = 25 Isolate the variable

Your turn: A cake taken out of the oven at temperature of 350 º F. It is placed on in a room with an ambient temperature of 70 º F to cool. Ten minutes later the temperature of the cake is 150 º F. When will the cake be cool enough to put the frosting on (90 º F) ? Calculate ‘k’ using the time to cool to the intermediate temperature. intermediate temperature. Once you have ‘k’, find ‘t’ for T(t) = desired temperature.

Your Turn: You are baking a cake. When you take the cake out of the oven, it is at 350ºF. The room temperature is 70ºF. The cooling rate is How many minutes will it take for the cake cool to 100ºF? Plug #’s into the formula. Isolate the power Undo the power Solve for ‘t’ t = 27.9 minutes t = 27.9 minutes

Measure of the acidity of a solution. Measure of the acidity of a solution. Acidity caused by the amount of hydrogen ion in solution.  concentration of the “hydronium” ion. Smallest detectable concentration: Maximum possible concentration: We convert this to pH. Max is 10 trillion times larger than smallest detectable!

pH In chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H + ]. The measure of acidity used is pH, the opposite of the common log of the hydrogen- ion concentration: pH = -log [H + ] More acidic solutions have higher hydrogen-ion concentrations and lower pH values. pH is the magitude (common log) of the hydrogen ion. The difference in pH between two solutions is the is the power of 10 difference between their hydrogen ion concentrations. of 10 difference between their hydrogen ion concentrations.

pH A solution has a pH of 2.3. What is the hydrogen-ion concentration? pH = -log [H+] pH = -log [H+]

Your Turn: 7. The pH of carbonated water is 3.9 and the pH of household ammonia is What are their hydrogen household ammonia is What are their hydrogen ion concentrations? ion concentrations? 8. For problem #9, how many times greater is the hydrogen ion concetration of carbonate water that hydrogen ion concetration of carbonate water that that of ammonia? that of ammonia? pH = -log [H + ] 9. By how many orders of magnitude do the concentrations differ? differ?

Real World Logarithmic Model Formula relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado. and the wind speed (s) inside the cone of the tornado. In 1925, a tornado traveled 220 miles through 3 states. Estimate the wind speed inside the tornado. (1) Plug numbers into the formula (2) Solve for the unknown variable in the formula

Your turn: Formula relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado. and the wind speed (s) inside the cone of the tornado. 10. A tornado traveled 100 miles. Estimate the wind speed inside the tornado. (1) Write formula (2) Plug numbers into the formula (3) Solve for the unknown variable in the formula

Real World Logarithmic Model What if the problem was: “The wind speed is 200mph, how far will the tornado travel on the ground? “The wind speed is 200mph, how far will the tornado travel on the ground? (1) Write formula (2) Plug numbers into the formula (3) Solve for the unknown variable in the formula (remember: “isolate the log, undo the log”)

Real World Logarithmic Model “The wind speed is 200mph, how far will the tornado travel on the ground? Subtract 65 from both sides Divide by 93 (both sides) remember: “isolate the radical” ?  same thing, “isolate the log” We need the “inverse function” of log (base 10). log (base 10). Exponential base 10

Your turn: Formula relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado. and the wind speed (s) inside the cone of the tornado. 11. Some storm chasers measured the speed of the wind inside a tornado. It was 275 mph. How far will the tornado travel along the ground?. (1) Plug numbers into the formula (2) Solve for the unknown variable in the formula

Population Growth Population (as a function of time) function of time) Initial population population Growth rate rate time It’s just a formula!!! The initial population of a colony of bacteria is The population increases by 50% is The population increases by 50% every hour. What is the population after 5 hours? every hour. What is the population after 5 hours? Percent rate of change (in decimal form) (in decimal form)

Simple Interest (savings account) Amount (as a function of time) function of time) Initial amount (“principle”) (“principle”)Interest rate rate time A bank account pays 3.5% interest per year. If you initially invest $200, how much money If you initially invest $200, how much money will you have after 5 years? will you have after 5 years?

What are logs used for in the real world? Anytime you need to measure a quantity that can range over a VERY BIG spread of values. Sound Intensity (watts/square meter) Threshold of hearing: Instant eardrum damage: We convert this to decibels.

Barking dog sound intensity: watts/sq meter watts/sq meter Sound Intensity Loudness of the sound (in decibels) as a function (in decibels) as a function of the sound intensity of the sound intensity Intensity of the sound in the sound in watts/sq meter watts/sq meter Intensity of sound at the threshold at the threshold of hearing ( of hearing ( watts per sq meter) watts per sq meter) How Loud is a dog’s bark? (in decibels)

An ambulance’s siren is 120 decibels loud. What is the sound intensity of the siren? Sound Intensity Substitute values from the problem into the formula. “Isolate the log” (divide by 10)

An ambulance’s siren is 120 decibels loud. What is the sound intensity of the siren? Sound Intensity “Expand the quotient” (quotient property of logarithms) Power property of logs. Simplify log10 Subtract 12 Apply inverse of log base 10

Your turn: 6. The front row of a rock concert has a sound intensity of The reference sound intensity What is the sound level in decibels on the front row of the rock concert?

Earthquake Magnitude

Earthquake Intensity Is measured by the amplitude of the vibration felt at the measuring station. felt at the measuring station. T: Period of the seismic wave (seconds) B: a “fudge factor” to account for weakening of the seismic wave from origin to where it is measured. seismic wave from origin to where it is measured. Amplitude is measured in

Comparing Orders of Magnitude of Earthquake Intensities 2010 Gugerat, India: R = Athens, GreeceR = 5.9 We want to compare the amplitudes.

Comparing Orders of Magnitude of Earthquake Intensities 100 times greater !! 7.9 – 5.9 is the difference in magnitudes (difference between the natural logs of two different numbers). between the natural logs of two different numbers). The difference in magnitudes is the power of 10 difference between the two different numbers). between the two different numbers).